Recursion, Induction and Rewriting

In this notebook, we're going to use Imandra to prove some interesting properties of functional programs. We'll learn a bit about induction, lemmas and rewrite rules along the way.

Let's start by defining our own append function on lists. It should take two lists as input, and compute the result of appending (concatenating) the elements of the first list with those of the second list.

Note: Imandra contains a List module with predefined functions like List.append (i.e., (@)), List.rev, etc. However, we define our own versions of these functions by hand in this notebook so that all definitions and proofs are illustrated from scratch.

In [1]:
let rec append x y =
 match x with
  | [] -> y
  | x :: xs -> x :: append xs y
Out[1]:
val append : 'a list -> 'a list -> 'a list = <fun>
termination proof

Termination proof

call `append (List.tl x) y` from `append x y`
original:append x y
sub:append (List.tl x) y
original ordinal:Ordinal.Int (_cnt x)
sub ordinal:Ordinal.Int (_cnt (List.tl x))
path:[x <> []]
proof:
detailed proof
ground_instances:3
definitions:0
inductions:0
search_time:
0.013s
details:
Expand
smt_stats:
num checks:8
arith assert lower:5
arith tableau max rows:4
arith tableau max columns:11
arith pivots:4
rlimit count:1806
mk clause:12
datatype occurs check:12
mk bool var:52
arith assert upper:5
datatype splits:1
decisions:10
arith row summations:7
propagations:12
conflicts:7
arith fixed eqs:4
datatype accessor ax:10
arith conflicts:1
arith num rows:4
datatype constructor ax:8
num allocs:5376899
final checks:4
added eqs:37
del clause:4
arith eq adapter:5
memory:15.190000
max memory:15.190000
Expand
  • start[0.013s]
      let (_x_0 : int) = count.list (const 0) x in
      let (_x_1 : ty_0 list) = List.tl x in
      let (_x_2 : int) = count.list (const 0) _x_1 in
      x <> [] && ((_x_0 >= 0) && (_x_2 >= 0))
      ==> not (_x_1 <> [])
          || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
  • simplify
    into:
    let (_x_0 : ty_0 list) = List.tl x in
    let (_x_1 : int) = count.list (const 0) _x_0 in
    let (_x_2 : int) = count.list (const 0) x in
    not (_x_0 <> []) || Ordinal.( << ) (Ordinal.Int _x_1) (Ordinal.Int _x_2)
    || not (x <> [] && (_x_2 >= 0) && (_x_1 >= 0))
    expansions:
    []
    rewrite_steps:
      forward_chaining:
      • unroll
        expr:
        (|Ordinal.<<| (|Ordinal.Int_79/boot|
                        (|count.list_395/server| (|get.::.1_376/server|…
        expansions:
        • unroll
          expr:
          (|count.list_395/server| (|get.::.1_376/server| x_382/server))
          expansions:
          • unroll
            expr:
            (|count.list_395/server| x_382/server)
            expansions:
            • Unsat

            Let's compute a bit with this function to understand how it works.

            In [2]:
            append [1;2;3] [4;5;6]
            
            Out[2]:
            - : Z.t list = [1;2;3;4;5;6]
            
            In [3]:
            append [] [4;5;6]
            
            Out[3]:
            - : Z.t list = [4;5;6]
            
            In [4]:
            append [1;2;3] []
            
            Out[4]:
            - : Z.t list = [1;2;3]
            

            We can ask Imandra some questions about append. For example, do there exist lists x and y such that append x y = [1;2;3;4;5;6;7;8]?

            In [5]:
            verify (fun x y -> append x y <> [1;2;3;4;5;6;7;8])
            
            Out[5]:
            - : Z.t list -> Z.t list -> bool = <fun>
            module CX : sig val x : Z.t list val y : Z.t list end
            
            Counterexample (after 1 steps, 0.013s):
            let x : int list = []
            let y : int list = [1; 2; 3; 4; 5; 6; 7; 8]
            
            Refuted
            proof attempt
            ground_instances:1
            definitions:0
            inductions:0
            search_time:
            0.013s
            details:
            Expand
            smt_stats:
            num checks:3
            rlimit count:619
            mk clause:3
            datatype occurs check:28
            mk bool var:29
            datatype splits:1
            decisions:2
            propagations:2
            conflicts:1
            datatype accessor ax:10
            datatype constructor ax:2
            num allocs:9770223
            final checks:3
            added eqs:26
            memory:15.430000
            max memory:15.430000
            Expand
            • start[0.013s] not (append ( :var_0: ) ( :var_1: ) = [1; 2; 3; 4; 5; 6; 7; 8])
            • unroll
              expr:
              (append_428/server x_1262/client y_1263/client)
              expansions:
              • Sat (Some let x : int list = [] let y : int list = [(Z.of_nativeint (1n)); (Z.of_nativeint (2n)); (Z.of_nativeint (3n)); (Z.of_nativeint (4n)); (Z.of_nativeint (5n)); (Z.of_nativeint (6n)); (Z.of_nativeint (7n)); (Z.of_nativeint (8n))] )

              Ah, of course! What if we make the problem a little harder -- perhaps we want x and y to be the same length:

              In [6]:
              verify (fun x y -> List.length x = List.length y ==> append x y <> [1;2;3;4;5;6;7;8])
              
              Out[6]:
              - : Z.t list -> Z.t list -> bool = <fun>
              module CX : sig val x : Z.t list val y : Z.t list end
              
              Counterexample (after 15 steps, 0.022s):
              let x : int list = [1; 2; 3; 4]
              let y : int list = [5; 6; 7; 8]
              
              Refuted
              proof attempt
              ground_instances:15
              definitions:0
              inductions:0
              search_time:
              0.022s
              details:
              Expand
              smt_stats:
              arith offset eqs:31
              num checks:31
              arith assert lower:106
              arith tableau max rows:18
              arith tableau max columns:38
              arith pivots:107
              rlimit count:15579
              mk clause:267
              datatype occurs check:229
              mk bool var:999
              arith assert upper:106
              datatype splits:13
              decisions:280
              arith row summations:251
              arith bound prop:2
              propagations:783
              interface eqs:2
              conflicts:63
              arith fixed eqs:139
              arith assume eqs:2
              datatype accessor ax:102
              minimized lits:25
              arith num rows:18
              arith assert diseq:1
              datatype constructor ax:191
              num allocs:15650939
              final checks:28
              added eqs:2112
              del clause:162
              arith eq adapter:167
              time:0.001000
              memory:16.040000
              max memory:16.040000
              Expand
              • start[0.022s]
                  List.length ( :var_0: ) = List.length ( :var_1: )
                  ==> not (append ( :var_0: ) ( :var_1: ) = [1; 2; 3; 4; 5; 6; 7; 8])
              • simplify

                into:
                not (List.length ( :var_0: ) = List.length ( :var_1: ))
                || not (append ( :var_0: ) ( :var_1: ) = [1; 2; 3; 4; 5; 6; 7; 8])
                expansions:
                []
                rewrite_steps:
                  forward_chaining:
                  • unroll
                    expr:
                    (append_445/server x_1265/client y_1266/client)
                    expansions:
                    • unroll
                      expr:
                      (|List.length_441/server| y_1266/client)
                      expansions:
                      • unroll
                        expr:
                        (|List.length_441/server| x_1265/client)
                        expansions:
                        • unroll
                          expr:
                          (append_445/server (|get.::.1_440/server| x_1265/client) y_1266/client)
                          expansions:
                          • unroll
                            expr:
                            (|List.length_441/server| (|get.::.1_440/server| y_1266/client))
                            expansions:
                            • unroll
                              expr:
                              (|List.length_441/server| (|get.::.1_440/server| x_1265/client))
                              expansions:
                              • unroll
                                expr:
                                (append_445/server
                                  (|get.::.1_440/server| (|get.::.1_440/server| x_1265/client))
                                  y_1266/client)
                                expansions:
                                • unroll
                                  expr:
                                  (|List.length_441/server|
                                    (|get.::.1_440/server| (|get.::.1_440/server| y_1266/client)))
                                  expansions:
                                  • unroll
                                    expr:
                                    (|List.length_441/server|
                                      (|get.::.1_440/server| (|get.::.1_440/server| x_1265/client)))
                                    expansions:
                                    • unroll
                                      expr:
                                      (append_445/server
                                        (|get.::.1_440/server|
                                          (|get.::.1_440/server| (|get.::.1_440/server| x_1265…
                                      expansions:
                                      • unroll
                                        expr:
                                        (|List.length_441/server|
                                          (|get.::.1_440/server|
                                            (|get.::.1_440/server| (|get.::.1_440/server|…
                                        expansions:
                                        • unroll
                                          expr:
                                          (|List.length_441/server|
                                            (|get.::.1_440/server|
                                              (|get.::.1_440/server| (|get.::.1_440/server|…
                                          expansions:
                                          • unroll
                                            expr:
                                            (let ((a!1 (|get.::.1_440/server|
                                                         (|get.::.1_440/server|
                                                           (|get.::.1_440/…
                                            expansions:
                                            • unroll
                                              expr:
                                              (let ((a!1 (|get.::.1_440/server|
                                                           (|get.::.1_440/server|
                                                             (|get.::.1_440/…
                                              expansions:
                                              • unroll
                                                expr:
                                                (let ((a!1 (|get.::.1_440/server|
                                                             (|get.::.1_440/server|
                                                               (|get.::.1_440/…
                                                expansions:
                                                • Sat (Some let x : int list = [(Z.of_nativeint (1n)); (Z.of_nativeint (2n)); (Z.of_nativeint (3n)); (Z.of_nativeint (4n))] let y : int list = [(Z.of_nativeint (5n)); (Z.of_nativeint (6n)); (Z.of_nativeint (7n)); (Z.of_nativeint (8n))] )

                                                Nice! Remember that counterexamples are always reflected into the runtime in a module called CX. So, we can compute with this counterexample and see that it indeed refutes our conjecture:

                                                In [7]:
                                                append CX.x CX.y
                                                
                                                Out[7]:
                                                - : Z.t list = [1;2;3;4;5;6;7;8]
                                                

                                                Let's now investigate a more interesting conjecture: That append is associative.

                                                That is,

                                                (forall (x,y,z) : 'a list. append x (append y z) = append (append x y) z).

                                                Is this property true? Let's ask Imandra.

                                                In [8]:
                                                verify (fun x y z -> append x (append y z) = append (append x y) z)
                                                
                                                Out[8]:
                                                - : 'a list -> 'a list -> 'a list -> bool = <fun>
                                                
                                                Unknown (Verified up to bound 100)
                                                Expand
                                                expanded:
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))) y
                                                • append (List.tl x) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl x))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))) (append y z)
                                                • append (List.tl (List.tl (append x y))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl x))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))) z
                                                • append x y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))) z
                                                • append (List.tl (List.tl (List.tl x))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))) y
                                                • append x (append y z)
                                                • append (append x y) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl x))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl y)))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) y
                                                • append (List.tl y) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (append x y))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (append x y)))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) (append y z)
                                                • append y z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))))))) y
                                                • append (List.tl (List.tl y)) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl y))))) z
                                                • append (List.tl (List.tl x)) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl y))) z
                                                • append (List.tl (List.tl (List.tl (List.tl x)))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))))) y
                                                • append (List.tl (List.tl x)) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))))) y
                                                • append (List.tl (append x y)) z
                                                • append (List.tl (List.tl (List.tl (List.tl x)))) (append y z)
                                                • append (List.tl x) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y))))))))))))))) z
                                                blocked:
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))))))))) y
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))))))))))))))))))) (append y z)
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl y)))))))))))))))))))) z
                                                • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))))))))))))))))))))) z
                                                proof attempt
                                                ground_instances:100
                                                definitions:0
                                                inductions:0
                                                search_time:
                                                0.145s
                                                Expand
                                                • start[0.145s]
                                                    append ( :var_0: ) (append ( :var_1: ) ( :var_2: ))
                                                    = append (append ( :var_0: ) ( :var_1: )) ( :var_2: )
                                                • unroll
                                                  expr:
                                                  (append_495/server (append_495/server x_488/server y_489/server) z_490/server)
                                                  expansions:
                                                  • unroll
                                                    expr:
                                                    (append_495/server x_488/server y_489/server)
                                                    expansions:
                                                    • unroll
                                                      expr:
                                                      (append_495/server x_488/server (append_495/server y_489/server z_490/server))
                                                      expansions:
                                                      • unroll
                                                        expr:
                                                        (append_495/server y_489/server z_490/server)
                                                        expansions:
                                                        • unroll
                                                          expr:
                                                          (append_495/server
                                                            (|get.::.1_487/server| (append_495/server x_488/server y_489/server))
                                                            z_490/s…
                                                          expansions:
                                                          • unroll
                                                            expr:
                                                            (append_495/server (|get.::.1_487/server| x_488/server) y_489/server)
                                                            expansions:
                                                            • unroll
                                                              expr:
                                                              (append_495/server
                                                                (|get.::.1_487/server| x_488/server)
                                                                (append_495/server y_489/server z_490/se…
                                                              expansions:
                                                              • unroll
                                                                expr:
                                                                (append_495/server (|get.::.1_487/server| y_489/server) z_490/server)
                                                                expansions:
                                                                • unroll
                                                                  expr:
                                                                  (append_495/server
                                                                    (|get.::.1_487/server|
                                                                      (|get.::.1_487/server| (append_495/server x_488/serv…
                                                                  expansions:
                                                                  • unroll
                                                                    expr:
                                                                    (append_495/server
                                                                      (|get.::.1_487/server| (|get.::.1_487/server| x_488/server))
                                                                      y_489/server)
                                                                    expansions:
                                                                    • unroll
                                                                      expr:
                                                                      (append_495/server
                                                                        (|get.::.1_487/server| (|get.::.1_487/server| x_488/server))
                                                                        (append_495/serv…
                                                                      expansions:
                                                                      • unroll
                                                                        expr:
                                                                        (append_495/server
                                                                          (|get.::.1_487/server|
                                                                            (|get.::.1_487/server| (|get.::.1_487/server| x_488/…
                                                                        expansions:
                                                                        • unroll
                                                                          expr:
                                                                          (append_495/server
                                                                            (|get.::.1_487/server|
                                                                              (|get.::.1_487/server| (|get.::.1_487/server| x_488/…
                                                                          expansions:
                                                                          • unroll
                                                                            expr:
                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                         (|get.::.1_487/server|
                                                                                           (|get.::.1_487/…
                                                                            expansions:
                                                                            • unroll
                                                                              expr:
                                                                              (append_495/server
                                                                                (|get.::.1_487/server| (|get.::.1_487/server| y_489/server))
                                                                                z_490/server)
                                                                              expansions:
                                                                              • unroll
                                                                                expr:
                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                             (|get.::.1_487/server|
                                                                                               (|get.::.1_487/…
                                                                                expansions:
                                                                                • unroll
                                                                                  expr:
                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                               (|get.::.1_487/server|
                                                                                                 (|get.::.1_487/…
                                                                                  expansions:
                                                                                  • unroll
                                                                                    expr:
                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                 (|get.::.1_487/server|
                                                                                                   (|get.::.1_487/…
                                                                                    expansions:
                                                                                    • unroll
                                                                                      expr:
                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                   (|get.::.1_487/server|
                                                                                                     (|get.::.1_487/…
                                                                                      expansions:
                                                                                      • unroll
                                                                                        expr:
                                                                                        (append_495/server
                                                                                          (|get.::.1_487/server|
                                                                                            (|get.::.1_487/server| (|get.::.1_487/server| y_489/…
                                                                                        expansions:
                                                                                        • unroll
                                                                                          expr:
                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                       (|get.::.1_487/server|
                                                                                                         (|get.::.1_487/…
                                                                                          expansions:
                                                                                          • unroll
                                                                                            expr:
                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                         (|get.::.1_487/server|
                                                                                                           (|get.::.1_487/…
                                                                                            expansions:
                                                                                            • unroll
                                                                                              expr:
                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                           (|get.::.1_487/server|
                                                                                                             (|get.::.1_487/…
                                                                                              expansions:
                                                                                              • unroll
                                                                                                expr:
                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                             (|get.::.1_487/server|
                                                                                                               (|get.::.1_487/…
                                                                                                expansions:
                                                                                                • unroll
                                                                                                  expr:
                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                               (|get.::.1_487/server|
                                                                                                                 (|get.::.1_487/…
                                                                                                  expansions:
                                                                                                  • unroll
                                                                                                    expr:
                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                 (|get.::.1_487/server|
                                                                                                                   (|get.::.1_487/…
                                                                                                    expansions:
                                                                                                    • unroll
                                                                                                      expr:
                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                   (|get.::.1_487/server|
                                                                                                                     (|get.::.1_487/…
                                                                                                      expansions:
                                                                                                      • unroll
                                                                                                        expr:
                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                     (|get.::.1_487/server|
                                                                                                                       (|get.::.1_487/…
                                                                                                        expansions:
                                                                                                        • unroll
                                                                                                          expr:
                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                       (|get.::.1_487/server|
                                                                                                                         (|get.::.1_487/…
                                                                                                          expansions:
                                                                                                          • unroll
                                                                                                            expr:
                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                         (|get.::.1_487/server|
                                                                                                                           (|get.::.1_487/…
                                                                                                            expansions:
                                                                                                            • unroll
                                                                                                              expr:
                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                           (|get.::.1_487/server|
                                                                                                                             (|get.::.1_487/…
                                                                                                              expansions:
                                                                                                              • unroll
                                                                                                                expr:
                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                             (|get.::.1_487/server|
                                                                                                                               (|get.::.1_487/…
                                                                                                                expansions:
                                                                                                                • unroll
                                                                                                                  expr:
                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                               (|get.::.1_487/server|
                                                                                                                                 (|get.::.1_487/…
                                                                                                                  expansions:
                                                                                                                  • unroll
                                                                                                                    expr:
                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                   (|get.::.1_487/…
                                                                                                                    expansions:
                                                                                                                    • unroll
                                                                                                                      expr:
                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                     (|get.::.1_487/…
                                                                                                                      expansions:
                                                                                                                      • unroll
                                                                                                                        expr:
                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                       (|get.::.1_487/…
                                                                                                                        expansions:
                                                                                                                        • unroll
                                                                                                                          expr:
                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                         (|get.::.1_487/…
                                                                                                                          expansions:
                                                                                                                          • unroll
                                                                                                                            expr:
                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                           (|get.::.1_487/…
                                                                                                                            expansions:
                                                                                                                            • unroll
                                                                                                                              expr:
                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                             (|get.::.1_487/…
                                                                                                                              expansions:
                                                                                                                              • unroll
                                                                                                                                expr:
                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                               (|get.::.1_487/…
                                                                                                                                expansions:
                                                                                                                                • unroll
                                                                                                                                  expr:
                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                  expansions:
                                                                                                                                  • unroll
                                                                                                                                    expr:
                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                    expansions:
                                                                                                                                    • unroll
                                                                                                                                      expr:
                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                      expansions:
                                                                                                                                      • unroll
                                                                                                                                        expr:
                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                        expansions:
                                                                                                                                        • unroll
                                                                                                                                          expr:
                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                          expansions:
                                                                                                                                          • unroll
                                                                                                                                            expr:
                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                            expansions:
                                                                                                                                            • unroll
                                                                                                                                              expr:
                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                              expansions:
                                                                                                                                              • unroll
                                                                                                                                                expr:
                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                expansions:
                                                                                                                                                • unroll
                                                                                                                                                  expr:
                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                  expansions:
                                                                                                                                                  • unroll
                                                                                                                                                    expr:
                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                    expansions:
                                                                                                                                                    • unroll
                                                                                                                                                      expr:
                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                      expansions:
                                                                                                                                                      • unroll
                                                                                                                                                        expr:
                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                        expansions:
                                                                                                                                                        • unroll
                                                                                                                                                          expr:
                                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                                          expansions:
                                                                                                                                                          • unroll
                                                                                                                                                            expr:
                                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                                            expansions:
                                                                                                                                                            • unroll
                                                                                                                                                              expr:
                                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                                              expansions:
                                                                                                                                                              • unroll
                                                                                                                                                                expr:
                                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                                expansions:
                                                                                                                                                                • unroll
                                                                                                                                                                  expr:
                                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                                  expansions:
                                                                                                                                                                  • unroll
                                                                                                                                                                    expr:
                                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                                    expansions:
                                                                                                                                                                    • unroll
                                                                                                                                                                      expr:
                                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                                      expansions:
                                                                                                                                                                      • unroll
                                                                                                                                                                        expr:
                                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                                        expansions:
                                                                                                                                                                        • unroll
                                                                                                                                                                          expr:
                                                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                                                          expansions:
                                                                                                                                                                          • unroll
                                                                                                                                                                            expr:
                                                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                                                            expansions:
                                                                                                                                                                            • unroll
                                                                                                                                                                              expr:
                                                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                                                              expansions:
                                                                                                                                                                              • unroll
                                                                                                                                                                                expr:
                                                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                                                expansions:
                                                                                                                                                                                • unroll
                                                                                                                                                                                  expr:
                                                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                                                  expansions:
                                                                                                                                                                                  • unroll
                                                                                                                                                                                    expr:
                                                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                                                    expansions:
                                                                                                                                                                                    • unroll
                                                                                                                                                                                      expr:
                                                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                                                      expansions:
                                                                                                                                                                                      • unroll
                                                                                                                                                                                        expr:
                                                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                                                        expansions:
                                                                                                                                                                                        • unroll
                                                                                                                                                                                          expr:
                                                                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                                                                          expansions:
                                                                                                                                                                                          • unroll
                                                                                                                                                                                            expr:
                                                                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                                                                            expansions:
                                                                                                                                                                                            • unroll
                                                                                                                                                                                              expr:
                                                                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                                                                              expansions:
                                                                                                                                                                                              • unroll
                                                                                                                                                                                                expr:
                                                                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                                                                expansions:
                                                                                                                                                                                                • unroll
                                                                                                                                                                                                  expr:
                                                                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                    expr:
                                                                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                      expr:
                                                                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                        expr:
                                                                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                          expr:
                                                                                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                            expr:
                                                                                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                              expr:
                                                                                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                       (|get.::.1_487/server|
                                                                                                                                                                                                                                                         (|get.::.1_487/…
                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                         (|get.::.1_487/server|
                                                                                                                                                                                                                                                           (|get.::.1_487/…
                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                           (|get.::.1_487/server|
                                                                                                                                                                                                                                                             (|get.::.1_487/…
                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                             (|get.::.1_487/server|
                                                                                                                                                                                                                                                               (|get.::.1_487/…
                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                               (|get.::.1_487/server|
                                                                                                                                                                                                                                                                 (|get.::.1_487/…
                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                                 (|get.::.1_487/server|
                                                                                                                                                                                                                                                                   (|get.::.1_487/…
                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                                   (|get.::.1_487/server|
                                                                                                                                                                                                                                                                     (|get.::.1_487/…
                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_487/server|
                                                                                                                                                                                                                                                                     (|get.::.1_487/server|
                                                                                                                                                                                                                                                                       (|get.::.1_487/…
                                                                                                                                                                                                                                                        expansions:

                                                                                                                                                                                                                                                        Imandra tells us that there are no counterexamples to this conjecture up to our current recursion unrolling bound (100).

                                                                                                                                                                                                                                                        This gives us some confidence that this conjecture is true. Let's now ask Imandra to prove it for all possible cases by induction.

                                                                                                                                                                                                                                                        In [9]:
                                                                                                                                                                                                                                                        verify (fun x y z -> append x (append y z) = append (append x y) z) [@@induct]
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Out[9]:
                                                                                                                                                                                                                                                        - : 'a list -> 'a list -> 'a list -> bool = <fun>
                                                                                                                                                                                                                                                        Goal:
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        append x (append y z) = append (append x y) z.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        1 nontautological subgoal.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Subgoal 1:
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                         append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Must try induction.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        The recursive terms in the conjecture suggest 3 inductions.
                                                                                                                                                                                                                                                        Subsumption and merging reduces this to 2.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Only 1 of those schemes are unflawed.
                                                                                                                                                                                                                                                        We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Induction scheme:
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                         (not (x <> []) ==> φ x y z) && (x <> [] && φ (List.tl x) y z ==> φ x y z).
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        2 nontautological subgoals.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Subgoal 1.2:
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                         C0. x <> []
                                                                                                                                                                                                                                                         C1. append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definition of append.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Subgoal 1.1:
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                         H0. x <> []
                                                                                                                                                                                                                                                         H1. append (List.tl x) (append y z) = append (append (List.tl x) y) z
                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                         append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definition of append.
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                         Rules:
                                                                                                                                                                                                                                                            (:def append)
                                                                                                                                                                                                                                                            (:induct append)
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                        Proved
                                                                                                                                                                                                                                                        proof
                                                                                                                                                                                                                                                        ground_instances:0
                                                                                                                                                                                                                                                        definitions:5
                                                                                                                                                                                                                                                        inductions:1
                                                                                                                                                                                                                                                        search_time:
                                                                                                                                                                                                                                                        0.087s
                                                                                                                                                                                                                                                        Expand
                                                                                                                                                                                                                                                        • start[0.087s, "Goal"]
                                                                                                                                                                                                                                                            append ( :var_0: ) (append ( :var_1: ) ( :var_2: ))
                                                                                                                                                                                                                                                            = append (append ( :var_0: ) ( :var_1: )) ( :var_2: )
                                                                                                                                                                                                                                                        • subproof

                                                                                                                                                                                                                                                          append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                          • start[0.087s, "1"] append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                          • induction on (functional ?)
                                                                                                                                                                                                                                                            :scheme (not (x <> []) ==> φ x y z)
                                                                                                                                                                                                                                                                    && (x <> [] && φ (List.tl x) y z ==> φ x y z)
                                                                                                                                                                                                                                                          • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                   let (_x_1 : sko_ty_0 list) = append y z in
                                                                                                                                                                                                                                                                   let (_x_2 : bool) = append x _x_1 = append (append x y) z in
                                                                                                                                                                                                                                                                   let (_x_3 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                   (_x_0 || _x_2)
                                                                                                                                                                                                                                                                   && (not (_x_0 && (append _x_3 _x_1 = append (append _x_3 y) z))
                                                                                                                                                                                                                                                                       || _x_2)
                                                                                                                                                                                                                                                                   :cases [x <> [] || (append x (append y z) = append (append x y) z);
                                                                                                                                                                                                                                                                           let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                           let (_x_1 : sko_ty_0 list) = append y z in
                                                                                                                                                                                                                                                                           not (x <> [])
                                                                                                                                                                                                                                                                           || not (append _x_0 _x_1 = append (append _x_0 y) z)
                                                                                                                                                                                                                                                                           || (append x _x_1 = append (append x y) z)])
                                                                                                                                                                                                                                                            • subproof
                                                                                                                                                                                                                                                              let (_x_0 : sko_ty_0 list) = List.tl x in let (_x_1 : sko_ty_0 list) = append y z in not (x <> []) || not (append _x_0 _x_1 = append (append _x_0 y) z) || (append x _x_1 = append (append x y) z)
                                                                                                                                                                                                                                                              • start[0.042s, "1.1"]
                                                                                                                                                                                                                                                                  let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                  let (_x_1 : sko_ty_0 list) = append y z in
                                                                                                                                                                                                                                                                  not (x <> []) || not (append _x_0 _x_1 = append (append _x_0 y) z)
                                                                                                                                                                                                                                                                  || (append x _x_1 = append (append x y) z)
                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                true
                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                [append, append, append]
                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                  forward_chaining:
                                                                                                                                                                                                                                                                • subproof
                                                                                                                                                                                                                                                                  x <> [] || (append x (append y z) = append (append x y) z)
                                                                                                                                                                                                                                                                  • start[0.042s, "1.2"]
                                                                                                                                                                                                                                                                      x <> [] || (append x (append y z) = append (append x y) z)
                                                                                                                                                                                                                                                                  • simplify
                                                                                                                                                                                                                                                                    into:
                                                                                                                                                                                                                                                                    true
                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                    [append, append]
                                                                                                                                                                                                                                                                    rewrite_steps:
                                                                                                                                                                                                                                                                      forward_chaining:

                                                                                                                                                                                                                                                                Beautiful! So, we see Imandra proved our goal by induction on x, using a structural induction principle derived from the 'a list datatype. We now know the property holds for all possible inputs (of which there are infinitely many!).

                                                                                                                                                                                                                                                                By the way, we can always view Imandra's current session configuration with the #config directive:

                                                                                                                                                                                                                                                                In [10]:
                                                                                                                                                                                                                                                                #config
                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                Out[10]:
                                                                                                                                                                                                                                                                ----------------------------------------------------------------------------
                                                                                                                                                                                                                                                                Imandra session configuration
                                                                                                                                                                                                                                                                ----------------------------------------------------------------------------
                                                                                                                                                                                                                                                                 0. Mode: logic (see: program,logic)
                                                                                                                                                                                                                                                                 1. Load path: [] (see: dir)
                                                                                                                                                                                                                                                                 2. Undo (implicit push): false (see: undo)
                                                                                                                                                                                                                                                                 3. Console tags: waterfall, suggestions (see: console_tags)
                                                                                                                                                                                                                                                                 4. Console color: true (see: console_color)
                                                                                                                                                                                                                                                                 5. Interactive hints: true (see: interactive_hints)
                                                                                                                                                                                                                                                                 6. Timeout: 60000 (see: timeout)
                                                                                                                                                                                                                                                                 7. Allow redefinition: true (see: redef)
                                                                                                                                                                                                                                                                 8. Recursion unroll depth: 100 (see: unroll)
                                                                                                                                                                                                                                                                 9. SAT seed: 0 (see: sat_seed)
                                                                                                                                                                                                                                                                10. SMT seed: 0 (see: smt_seed)
                                                                                                                                                                                                                                                                11. Solver seed: 0 (see: solver_seed)
                                                                                                                                                                                                                                                                12. Top results: true (see: top_results)
                                                                                                                                                                                                                                                                13. Validate definitions: true (see: validate)
                                                                                                                                                                                                                                                                14. Skip proofs: false (see: skip_proofs)
                                                                                                                                                                                                                                                                15. Max induction depth: 3 (see: max_induct)
                                                                                                                                                                                                                                                                16. Induction unroll depth: 8 (see: induct_unroll)
                                                                                                                                                                                                                                                                17. Backchain limit: 100 (see: backchain_limit)
                                                                                                                                                                                                                                                                18. Enable all definitions when unrolling: true (see: unroll_enable_all)
                                                                                                                                                                                                                                                                19. Algebraic number approx. precision: 10 (see: pp_approx)
                                                                                                                                                                                                                                                                20. Reflect approximate models: false (see: reflect_approx_models)
                                                                                                                                                                                                                                                                21. Max size for callgraphs: 80 (see: callgraph_size)
                                                                                                                                                                                                                                                                22. Include expansion steps in callgraphs: false (see: callgraph_include_expansions)
                                                                                                                                                                                                                                                                23. Store proofs: false (see: store_proofs)
                                                                                                                                                                                                                                                                ----------------------------------------------------------------------------
                                                                                                                                                                                                                                                                

                                                                                                                                                                                                                                                                Lemmas and Rules

                                                                                                                                                                                                                                                                When we use the verify command, we give it a closed formula (e.g., a lambda term such as (fun x y z -> foo x y z) and ask Imandra to prove that the goal always evaluates to true. Notice that the verify command does not give the verification goal a name.

                                                                                                                                                                                                                                                                Once we've proved a goal, we often want to record it as a theorem. This is both to document our verification progress and to make it possible for our proved goal to be used as a lemma in subsequent verification efforts.

                                                                                                                                                                                                                                                                This can be done by the theorem command.

                                                                                                                                                                                                                                                                For example, we could name our result assoc_append and install it as a theorem as follows:

                                                                                                                                                                                                                                                                theorem assoc_append x y z = append x (append y z) = append (append x y) z) [@@induct]
                                                                                                                                                                                                                                                                

                                                                                                                                                                                                                                                                By default, a theorem is not installed as a rule that will be used automatically in subsequent verification efforts. To install the theorem as a rule, we can use the [@@rewrite] or [@@forward_chaining] attributes.

                                                                                                                                                                                                                                                                Let's focus in this notebook on the use of theorems as rewrite rules.

                                                                                                                                                                                                                                                                Rewrite Rules

                                                                                                                                                                                                                                                                A rewrite rule is a theorem of the form (H1 && ... && Hk ==> LHS = RHS). It instructs Imandra's simplifier to replace terms it matches with the LHS with the suitably instantiated RHS, provided that the corresponding instantiations of the hypotheses H1, ..., Hk can be proved.

                                                                                                                                                                                                                                                                It is also allowed for the conclusion of the rule to be a Boolean term, e.g., (H1 && ... && Hk ==> foo) is interpreted with an RHS of true, i.e., (H1 && ... && Hk ==> foo = true).

                                                                                                                                                                                                                                                                Good rewrite rules can have a powerful normalising effect on formulas. We usually want to orient them so that the RHS is a better (i.e., simpler or more canonical) term than the LHS.

                                                                                                                                                                                                                                                                Let us see the use of rewrite rules through a prove about the function reverse.

                                                                                                                                                                                                                                                                Reverse

                                                                                                                                                                                                                                                                Let's define a function to reverse a list. Note how it uses our append function we defined above:

                                                                                                                                                                                                                                                                In [11]:
                                                                                                                                                                                                                                                                let rec reverse x =
                                                                                                                                                                                                                                                                 match x with
                                                                                                                                                                                                                                                                  | [] -> []
                                                                                                                                                                                                                                                                  | x :: xs -> append (reverse xs) [x]
                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                Out[11]:
                                                                                                                                                                                                                                                                val reverse : 'a list -> 'a list = <fun>
                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                termination proof

                                                                                                                                                                                                                                                                Termination proof

                                                                                                                                                                                                                                                                call `reverse (List.tl x)` from `reverse x`
                                                                                                                                                                                                                                                                original:reverse x
                                                                                                                                                                                                                                                                sub:reverse (List.tl x)
                                                                                                                                                                                                                                                                original ordinal:Ordinal.Int (_cnt x)
                                                                                                                                                                                                                                                                sub ordinal:Ordinal.Int (_cnt (List.tl x))
                                                                                                                                                                                                                                                                path:[x <> []]
                                                                                                                                                                                                                                                                proof:
                                                                                                                                                                                                                                                                detailed proof
                                                                                                                                                                                                                                                                ground_instances:3
                                                                                                                                                                                                                                                                definitions:0
                                                                                                                                                                                                                                                                inductions:0
                                                                                                                                                                                                                                                                search_time:
                                                                                                                                                                                                                                                                0.011s
                                                                                                                                                                                                                                                                details:
                                                                                                                                                                                                                                                                Expand
                                                                                                                                                                                                                                                                smt_stats:
                                                                                                                                                                                                                                                                num checks:8
                                                                                                                                                                                                                                                                arith assert lower:5
                                                                                                                                                                                                                                                                arith tableau max rows:4
                                                                                                                                                                                                                                                                arith tableau max columns:11
                                                                                                                                                                                                                                                                arith pivots:4
                                                                                                                                                                                                                                                                rlimit count:1806
                                                                                                                                                                                                                                                                mk clause:12
                                                                                                                                                                                                                                                                datatype occurs check:12
                                                                                                                                                                                                                                                                mk bool var:52
                                                                                                                                                                                                                                                                arith assert upper:5
                                                                                                                                                                                                                                                                datatype splits:1
                                                                                                                                                                                                                                                                decisions:10
                                                                                                                                                                                                                                                                arith row summations:7
                                                                                                                                                                                                                                                                propagations:12
                                                                                                                                                                                                                                                                conflicts:7
                                                                                                                                                                                                                                                                arith fixed eqs:4
                                                                                                                                                                                                                                                                datatype accessor ax:10
                                                                                                                                                                                                                                                                arith conflicts:1
                                                                                                                                                                                                                                                                arith num rows:4
                                                                                                                                                                                                                                                                datatype constructor ax:8
                                                                                                                                                                                                                                                                num allocs:195987902
                                                                                                                                                                                                                                                                final checks:4
                                                                                                                                                                                                                                                                added eqs:37
                                                                                                                                                                                                                                                                del clause:4
                                                                                                                                                                                                                                                                arith eq adapter:5
                                                                                                                                                                                                                                                                memory:17.760000
                                                                                                                                                                                                                                                                max memory:18.230000
                                                                                                                                                                                                                                                                Expand
                                                                                                                                                                                                                                                                • start[0.011s]
                                                                                                                                                                                                                                                                    let (_x_0 : int) = count.list (const 0) x in
                                                                                                                                                                                                                                                                    let (_x_1 : ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                    let (_x_2 : int) = count.list (const 0) _x_1 in
                                                                                                                                                                                                                                                                    x <> [] && ((_x_0 >= 0) && (_x_2 >= 0))
                                                                                                                                                                                                                                                                    ==> not (_x_1 <> [])
                                                                                                                                                                                                                                                                        || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
                                                                                                                                                                                                                                                                • simplify
                                                                                                                                                                                                                                                                  into:
                                                                                                                                                                                                                                                                  let (_x_0 : ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                  let (_x_1 : int) = count.list (const 0) _x_0 in
                                                                                                                                                                                                                                                                  let (_x_2 : int) = count.list (const 0) x in
                                                                                                                                                                                                                                                                  not (_x_0 <> []) || Ordinal.( << ) (Ordinal.Int _x_1) (Ordinal.Int _x_2)
                                                                                                                                                                                                                                                                  || not (x <> [] && (_x_2 >= 0) && (_x_1 >= 0))
                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                  []
                                                                                                                                                                                                                                                                  rewrite_steps:
                                                                                                                                                                                                                                                                    forward_chaining:
                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                      (|Ordinal.<<| (|Ordinal.Int_79/boot|
                                                                                                                                                                                                                                                                                      (|count.list_742/server| (|get.::.1_721/server|…
                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                        (|count.list_742/server| (|get.::.1_721/server| x_731/server))
                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                          (|count.list_742/server| x_731/server)
                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                          • Unsat

                                                                                                                                                                                                                                                                          Let's compute with reverse to help gain confidence we defined it correctly.

                                                                                                                                                                                                                                                                          In [12]:
                                                                                                                                                                                                                                                                          reverse [1;2;3;4;5]
                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                          Out[12]:
                                                                                                                                                                                                                                                                          - : Z.t list = [5;4;3;2;1]
                                                                                                                                                                                                                                                                          

                                                                                                                                                                                                                                                                          Let's ask Imandra, are there any lists that are their own reverse?

                                                                                                                                                                                                                                                                          In [13]:
                                                                                                                                                                                                                                                                          instance (fun x -> reverse x = x)
                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                          Out[13]:
                                                                                                                                                                                                                                                                          - : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                          module CX : sig type t = C_t_0 val x : t list end
                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                          Instance (after 1 steps, 0.009s):
                                                                                                                                                                                                                                                                          type t = C_t_0
                                                                                                                                                                                                                                                                          let x : t list = []
                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                          Instance
                                                                                                                                                                                                                                                                          proof attempt
                                                                                                                                                                                                                                                                          ground_instances:1
                                                                                                                                                                                                                                                                          definitions:0
                                                                                                                                                                                                                                                                          inductions:0
                                                                                                                                                                                                                                                                          search_time:
                                                                                                                                                                                                                                                                          0.009s
                                                                                                                                                                                                                                                                          details:
                                                                                                                                                                                                                                                                          Expand
                                                                                                                                                                                                                                                                          smt_stats:
                                                                                                                                                                                                                                                                          num checks:3
                                                                                                                                                                                                                                                                          rlimit count:423
                                                                                                                                                                                                                                                                          mk clause:4
                                                                                                                                                                                                                                                                          datatype occurs check:2
                                                                                                                                                                                                                                                                          mk bool var:12
                                                                                                                                                                                                                                                                          decisions:1
                                                                                                                                                                                                                                                                          propagations:2
                                                                                                                                                                                                                                                                          conflicts:1
                                                                                                                                                                                                                                                                          datatype accessor ax:2
                                                                                                                                                                                                                                                                          datatype constructor ax:1
                                                                                                                                                                                                                                                                          num allocs:223993232
                                                                                                                                                                                                                                                                          final checks:2
                                                                                                                                                                                                                                                                          added eqs:9
                                                                                                                                                                                                                                                                          memory:18.000000
                                                                                                                                                                                                                                                                          max memory:18.230000
                                                                                                                                                                                                                                                                          Expand
                                                                                                                                                                                                                                                                          • start[0.009s] reverse ( :var_0: ) = ( :var_0: )
                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                            (reverse_795/server x_792/server)
                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                            • Sat (Some type t = C_t_0 let x : t list = [] )

                                                                                                                                                                                                                                                                            Ah! Of course, the empty list is its own reverse. What about longer lists, such as those of length 5?

                                                                                                                                                                                                                                                                            Note that here we use the keyword instance, which tries to find values that satisfy the given property, instead of trying to prove the property.

                                                                                                                                                                                                                                                                            In [14]:
                                                                                                                                                                                                                                                                            instance (fun x -> List.length x >= 5 && reverse x = x)
                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                            Out[14]:
                                                                                                                                                                                                                                                                            - : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                            module CX :
                                                                                                                                                                                                                                                                              sig
                                                                                                                                                                                                                                                                                type t =
                                                                                                                                                                                                                                                                                    C_t_0
                                                                                                                                                                                                                                                                                  | C_t_1
                                                                                                                                                                                                                                                                                  | C_t_2
                                                                                                                                                                                                                                                                                  | C_t_3
                                                                                                                                                                                                                                                                                  | C_t_4
                                                                                                                                                                                                                                                                                  | C_t_5
                                                                                                                                                                                                                                                                                  | C_t_6
                                                                                                                                                                                                                                                                                  | C_t_7
                                                                                                                                                                                                                                                                                  | C_t_8
                                                                                                                                                                                                                                                                                  | C_t_9
                                                                                                                                                                                                                                                                                val x : t list
                                                                                                                                                                                                                                                                              end
                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                            Instance (after 32 steps, 0.034s):
                                                                                                                                                                                                                                                                            type t =
                                                                                                                                                                                                                                                                              C_t_0 | C_t_1 | C_t_2 | C_t_3 | C_t_4 | C_t_5 | C_t_6 | C_t_7 | C_t_8 | C_t_9
                                                                                                                                                                                                                                                                            let x : t list = [C_t_0; C_t_9; C_t_2; C_t_9; C_t_0]
                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                            Instance
                                                                                                                                                                                                                                                                            proof attempt
                                                                                                                                                                                                                                                                            ground_instances:32
                                                                                                                                                                                                                                                                            definitions:0
                                                                                                                                                                                                                                                                            inductions:0
                                                                                                                                                                                                                                                                            search_time:
                                                                                                                                                                                                                                                                            0.034s
                                                                                                                                                                                                                                                                            details:
                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                            smt_stats:
                                                                                                                                                                                                                                                                            num checks:65
                                                                                                                                                                                                                                                                            arith assert lower:52
                                                                                                                                                                                                                                                                            arith tableau max rows:14
                                                                                                                                                                                                                                                                            arith tableau max columns:25
                                                                                                                                                                                                                                                                            arith pivots:38
                                                                                                                                                                                                                                                                            rlimit count:32935
                                                                                                                                                                                                                                                                            mk clause:185
                                                                                                                                                                                                                                                                            datatype occurs check:783
                                                                                                                                                                                                                                                                            mk bool var:1459
                                                                                                                                                                                                                                                                            arith assert upper:41
                                                                                                                                                                                                                                                                            datatype splits:148
                                                                                                                                                                                                                                                                            decisions:1023
                                                                                                                                                                                                                                                                            arith row summations:212
                                                                                                                                                                                                                                                                            propagations:1029
                                                                                                                                                                                                                                                                            conflicts:85
                                                                                                                                                                                                                                                                            arith fixed eqs:40
                                                                                                                                                                                                                                                                            datatype accessor ax:204
                                                                                                                                                                                                                                                                            minimized lits:35
                                                                                                                                                                                                                                                                            arith conflicts:8
                                                                                                                                                                                                                                                                            arith num rows:14
                                                                                                                                                                                                                                                                            arith assert diseq:10
                                                                                                                                                                                                                                                                            datatype constructor ax:462
                                                                                                                                                                                                                                                                            num allocs:256383033
                                                                                                                                                                                                                                                                            final checks:76
                                                                                                                                                                                                                                                                            added eqs:3274
                                                                                                                                                                                                                                                                            del clause:111
                                                                                                                                                                                                                                                                            arith eq adapter:56
                                                                                                                                                                                                                                                                            memory:18.950000
                                                                                                                                                                                                                                                                            max memory:18.950000
                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                            • start[0.034s]
                                                                                                                                                                                                                                                                                (List.length ( :var_0: ) >= 5) && (reverse ( :var_0: ) = ( :var_0: ))
                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                              (reverse_841/server x_834/server)
                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                (|List.length_837/server| x_834/server)
                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                  (append_845/server
                                                                                                                                                                                                                                                                                    (reverse_841/server (|get.::.1_833/server| x_834/server))
                                                                                                                                                                                                                                                                                    (|::| (|get.::.0_83…
                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                    (reverse_841/server (|get.::.1_833/server| x_834/server))
                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                      (|List.length_837/server| (|get.::.1_833/server| x_834/server))
                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                        (append_845/server
                                                                                                                                                                                                                                                                                          (reverse_841/server
                                                                                                                                                                                                                                                                                            (|get.::.1_833/server| (|get.::.1_833/server| x_834/ser…
                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                          (reverse_841/server
                                                                                                                                                                                                                                                                                            (|get.::.1_833/server| (|get.::.1_833/server| x_834/server)))
                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                            (|List.length_837/server|
                                                                                                                                                                                                                                                                                              (|get.::.1_833/server| (|get.::.1_833/server| x_834/server)))
                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                              (append_845/server
                                                                                                                                                                                                                                                                                                (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                  (reverse_841/server (|get.::.1_833/server| x_834/ser…
                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                             (reverse_841/server
                                                                                                                                                                                                                                                                                                               (|get.::.1_833/ser…
                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                  (let ((a!1 (reverse_841/server
                                                                                                                                                                                                                                                                                                               (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                 (|get.::.1_833/ser…
                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                    (reverse_841/server
                                                                                                                                                                                                                                                                                                      (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                        (|get.::.1_833/server| (|get.::.1_833/server| x_834…
                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                      (|List.length_837/server|
                                                                                                                                                                                                                                                                                                        (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                          (|get.::.1_833/server| (|get.::.1_833/server|…
                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                     (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                       (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                       (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                         (reverse_841/se…
                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                         (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                           (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                           (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                             (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_841/server
                                                                                                                                                                                                                                                                                                                             (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                               (|get.::.1_833/ser…
                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                               (reverse_841/server
                                                                                                                                                                                                                                                                                                                                 (|get.::.1_833/ser…
                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                   (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                   (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                     (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                     (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                       (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                       (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                         (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                         (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                           (reverse_841/se…
                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_841/server
                                                                                                                                                                                                                                                                                                                                           (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                             (|get.::.1_833/ser…
                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                             (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                               (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                               (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                   (reverse_841/server
                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_833/ser…
                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_833/…
                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_833/server|
                                                                                                                                                                                                                                                                                                                                                           (reverse_841/se…
                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                            • Sat (Some type t = C_t_0 | C_t_1 | C_t_2 | C_t_3 | C_t_4 | C_t_5 | C_t_6 | C_t_7 | C_t_8 | C_t_9 let x : t list = [C_t_0; C_t_9; C_t_2; C_t_9; C_t_0] )

                                                                                                                                                                                                                                                                                                                                            Ah! Of course, palindromes! This counterexample is also interesting because, in addition to giving us a concrete value CX.x, this counterexample also involves the synthesis of an algebraic datatype. This is because our goal was polymorphic. If we want, for example, an int list counterexample, we can simply annotate our goal with types:

                                                                                                                                                                                                                                                                                                                                            In [15]:
                                                                                                                                                                                                                                                                                                                                            instance (fun (x : int list) -> List.length x >= 5 && reverse x = x)
                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                            Out[15]:
                                                                                                                                                                                                                                                                                                                                            - : Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                            module CX : sig val x : Z.t list end
                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                            Instance (after 32 steps, 0.035s):
                                                                                                                                                                                                                                                                                                                                            let x : int list = [6; 10; 8; 10; 6]
                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                            Instance
                                                                                                                                                                                                                                                                                                                                            proof attempt
                                                                                                                                                                                                                                                                                                                                            ground_instances:32
                                                                                                                                                                                                                                                                                                                                            definitions:0
                                                                                                                                                                                                                                                                                                                                            inductions:0
                                                                                                                                                                                                                                                                                                                                            search_time:
                                                                                                                                                                                                                                                                                                                                            0.035s
                                                                                                                                                                                                                                                                                                                                            details:
                                                                                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                                                                                            smt_stats:
                                                                                                                                                                                                                                                                                                                                            num checks:65
                                                                                                                                                                                                                                                                                                                                            arith assert lower:52
                                                                                                                                                                                                                                                                                                                                            arith tableau max rows:14
                                                                                                                                                                                                                                                                                                                                            arith tableau max columns:25
                                                                                                                                                                                                                                                                                                                                            arith pivots:38
                                                                                                                                                                                                                                                                                                                                            rlimit count:32925
                                                                                                                                                                                                                                                                                                                                            mk clause:185
                                                                                                                                                                                                                                                                                                                                            datatype occurs check:783
                                                                                                                                                                                                                                                                                                                                            mk bool var:1459
                                                                                                                                                                                                                                                                                                                                            arith assert upper:41
                                                                                                                                                                                                                                                                                                                                            datatype splits:148
                                                                                                                                                                                                                                                                                                                                            decisions:1023
                                                                                                                                                                                                                                                                                                                                            arith row summations:212
                                                                                                                                                                                                                                                                                                                                            propagations:1029
                                                                                                                                                                                                                                                                                                                                            conflicts:85
                                                                                                                                                                                                                                                                                                                                            arith fixed eqs:40
                                                                                                                                                                                                                                                                                                                                            datatype accessor ax:204
                                                                                                                                                                                                                                                                                                                                            minimized lits:35
                                                                                                                                                                                                                                                                                                                                            arith conflicts:8
                                                                                                                                                                                                                                                                                                                                            arith num rows:14
                                                                                                                                                                                                                                                                                                                                            arith assert diseq:10
                                                                                                                                                                                                                                                                                                                                            datatype constructor ax:462
                                                                                                                                                                                                                                                                                                                                            num allocs:294402113
                                                                                                                                                                                                                                                                                                                                            final checks:76
                                                                                                                                                                                                                                                                                                                                            added eqs:3274
                                                                                                                                                                                                                                                                                                                                            del clause:111
                                                                                                                                                                                                                                                                                                                                            arith eq adapter:56
                                                                                                                                                                                                                                                                                                                                            memory:19.670000
                                                                                                                                                                                                                                                                                                                                            max memory:19.670000
                                                                                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                                                                                            • start[0.035s]
                                                                                                                                                                                                                                                                                                                                                (List.length ( :var_0: ) >= 5) && (reverse ( :var_0: ) = ( :var_0: ))
                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                              (reverse_957/server x_1289/client)
                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                (|List.length_953/server| x_1289/client)
                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                  (append_961/server
                                                                                                                                                                                                                                                                                                                                                    (reverse_957/server (|get.::.1_952/server| x_1289/client))
                                                                                                                                                                                                                                                                                                                                                    (|::| (|get.::.0_9…
                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                    (reverse_957/server (|get.::.1_952/server| x_1289/client))
                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                      (|List.length_953/server| (|get.::.1_952/server| x_1289/client))
                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                        (append_961/server
                                                                                                                                                                                                                                                                                                                                                          (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_952/server| (|get.::.1_952/server| x_1289/cl…
                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                          (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_952/server| (|get.::.1_952/server| x_1289/client)))
                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                            (|List.length_953/server|
                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_952/server| (|get.::.1_952/server| x_1289/client)))
                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                              (append_961/server
                                                                                                                                                                                                                                                                                                                                                                (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                  (reverse_957/server (|get.::.1_952/server| x_1289/cl…
                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                             (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_952/ser…
                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_952/ser…
                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                    (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_952/server| (|get.::.1_952/server| x_128…
                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                      (|List.length_953/server|
                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                          (|get.::.1_952/server| (|get.::.1_952/server|…
                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                         (reverse_957/se…
                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_952/ser…
                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                               (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_952/ser…
                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                           (reverse_957/se…
                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_952/ser…
                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                   (reverse_957/server
                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_952/ser…
                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_952/…
                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_952/server|
                                                                                                                                                                                                                                                                                                                                                                                                                           (reverse_957/se…
                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                            • Sat (Some let x : int list = [(Z.of_nativeint (6n)); (Z.of_nativeint (10n)); (Z.of_nativeint (8n)); (Z.of_nativeint (10n)); (Z.of_nativeint (6n))] )

                                                                                                                                                                                                                                                                                                                                                                                                            Excellent! And we can of course compute with our counterexample:

                                                                                                                                                                                                                                                                                                                                                                                                            In [16]:
                                                                                                                                                                                                                                                                                                                                                                                                            reverse CX.x
                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                            Out[16]:
                                                                                                                                                                                                                                                                                                                                                                                                            - : Z.t list = [6;10;8;10;6]
                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                            In [17]:
                                                                                                                                                                                                                                                                                                                                                                                                            reverse CX.x = CX.x
                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                            Out[17]:
                                                                                                                                                                                                                                                                                                                                                                                                            - : bool = true
                                                                                                                                                                                                                                                                                                                                                                                                            

                                                                                                                                                                                                                                                                                                                                                                                                            Reverse of Reverse is the Identity

                                                                                                                                                                                                                                                                                                                                                                                                            Now that we've defined our reverse function and experimented a bit with it, let's try to prove an interesting theorem. Let's prove that

                                                                                                                                                                                                                                                                                                                                                                                                            (forall x, (reverse (reverse x)) = x).

                                                                                                                                                                                                                                                                                                                                                                                                            That is, if we take an arbitrary list x and we reverse it twice, we always get our original x back unscathed.

                                                                                                                                                                                                                                                                                                                                                                                                            As usual, let's start by asking Imandra to verify this via bounded checking (a.k.a. recursive unrolling).

                                                                                                                                                                                                                                                                                                                                                                                                            In [18]:
                                                                                                                                                                                                                                                                                                                                                                                                            verify (fun x -> reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                            Out[18]:
                                                                                                                                                                                                                                                                                                                                                                                                            - : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                            Unknown (Verified up to bound 100)
                                                                                                                                                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                                                                                                                                                            expanded:
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse x
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (reverse x)))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x)))))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (reverse x))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (reverse x)))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (reverse x)))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x))))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (reverse x))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x)))))))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (reverse x))) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (reverse (List.tl x)) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl x)))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (reverse x)))))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x))))))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (reverse x))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl x))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (reverse x))))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x)))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (reverse (List.tl x)))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x)))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x)))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (reverse (List.tl x))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (reverse x)))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl x))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x)))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (reverse x))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x)))))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (reverse x)))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (reverse (List.tl x))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            blocked:
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (reverse x) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))))))))
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl x in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = reverse x in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (reverse x)))) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x)))))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x))))))))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (reverse x))) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x))))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (reverse x)) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            • let (_x_0 : sko_ty_0 list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse x)))))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                            proof attempt
                                                                                                                                                                                                                                                                                                                                                                                                            ground_instances:100
                                                                                                                                                                                                                                                                                                                                                                                                            definitions:0
                                                                                                                                                                                                                                                                                                                                                                                                            inductions:0
                                                                                                                                                                                                                                                                                                                                                                                                            search_time:
                                                                                                                                                                                                                                                                                                                                                                                                            0.223s
                                                                                                                                                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                                                                                                                                                            • start[0.223s] reverse (reverse ( :var_0: )) = ( :var_0: )
                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                              (reverse_1086/server (reverse_1086/server x_1083/server))
                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_1086/server x_1083/server)
                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                  (append_1090/server
                                                                                                                                                                                                                                                                                                                                                                                                                    (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_1082/server| (reverse_1086/server x_1083/s…
                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                    (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_1082/server| (reverse_1086/server x_1083/server)))
                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                      (append_1090/server
                                                                                                                                                                                                                                                                                                                                                                                                                        (reverse_1086/server (|get.::.1_1082/server| x_1083/server))
                                                                                                                                                                                                                                                                                                                                                                                                                        (|::| (|get.::.…
                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                        (reverse_1086/server (|get.::.1_1082/server| x_1083/server))
                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                            (append_1090/server
                                                                                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_1086/server (|get.::.1_1082/server| x_108…
                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                              (append_1090/server
                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_1082/server| (|get.::.1_1082/server| x_108…
                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                    (|get.::.1_1082/server| (|get.::.1_1082/server| x_1083/server)))
                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                    (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_1082/server| (reverse_1086/server x_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                   (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                    (|get.::.1_1082/server| (|get.::.1_1082/server| x…
                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_1086…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_10…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (reverse_1086/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1082/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1082/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Imandra tells us that there are no counterexamples up to our current unrolling bound (100).

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    So, our conjecture seems like a good candidate for proof by induction.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    In [19]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    verify (fun x -> reverse (reverse x) = x) [@@induct]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Out[19]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    - : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    reverse (reverse x) = x.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    We shall induct according to a scheme derived from reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (not (x <> []) ==> φ x) && (x <> [] && φ (List.tl x) ==> φ x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. reverse (reverse (List.tl x)) = List.tl x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definition of reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. reverse (reverse (List.tl x)) = List.tl x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (append (reverse (List.tl x)) [List.hd x]) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    We can eliminate destructors by the following substitution:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     x -> x1 :: x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1'':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. reverse (reverse x2) = x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (append (reverse x2) [x1]) = x1 :: x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Cross-fertilizing with:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (reverse x2) = x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1''':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (append (reverse x2) [x1]) = x1 :: (reverse (reverse x2))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Candidates for generalization:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1'''':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (append gen_1 [x1]) = x1 :: (reverse gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    The recursive terms in the conjecture suggest 2 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subsumption and merging reduces this to 1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (not (gen_1 <> []) ==> φ gen_1 x1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     && (gen_1 <> [] && φ (List.tl gen_1) x1 ==> φ gen_1 x1).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1''''.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. gen_1 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. reverse (append gen_1 [x1]) = x1 :: (reverse gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definitions of append and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1''''.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. gen_1 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. reverse (append (List.tl gen_1) [x1]) = x1 :: (reverse (List.tl gen_1))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (append gen_1 [x1]) = x1 :: (reverse gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of append and reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1''''.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. gen_1 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. reverse (append (List.tl gen_1) [x1]) = x1 :: (reverse (List.tl gen_1))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     append (reverse (append (List.tl gen_1) [x1])) [List.hd gen_1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = x1 :: (append (reverse (List.tl gen_1)) [List.hd gen_1])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But we verify Subgoal 1.1''''.1' by recursive unrolling.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:induct reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ground_instances:3
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    definitions:11
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    inductions:2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    0.505s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    details:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    smt_stats:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    num checks:7
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith-make-feasible:4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith-max-columns:4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    rlimit count:92948
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    mk clause:49
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype occurs check:33
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    mk bool var:93
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype splits:7
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    decisions:22
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    propagations:34
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    conflicts:4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype accessor ax:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype constructor ax:11
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    num allocs:2389740728
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    final checks:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    added eqs:108
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    del clause:36
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    memory:22.580000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    max memory:22.960000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • start[0.505s, "Goal"] reverse (reverse ( :var_0: )) = ( :var_0: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.505s, "1"] reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        :scheme (not (x <> []) ==> φ x) && (x <> [] && φ (List.tl x) ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_1 : bool) = reverse (reverse x) = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_2 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (_x_0 || _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               && (not (_x_0 && (reverse (reverse _x_2) = _x_2)) || _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               :cases [x <> [] || (reverse (reverse x) = x);
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       not (x <> []) || not (reverse (reverse _x_0) = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || (reverse (reverse x) = x)])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : sko_ty_0 list) = List.tl x in not (x <> []) || not (reverse (reverse _x_0) = _x_0) || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.449s, "1.1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              not (x <> []) || not (reverse (reverse _x_0) = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_1 : sko_ty_0 list) = reverse _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            not (x <> []) || not (reverse _x_1 = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            || (reverse (append _x_1 [List.hd x]) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Elim_destructor (:cstor ( :: ) :replace x1 :: x2 :context [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Generalize (let (_x_0 : sko_ty_0 list) = reverse x2 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse (append _x_0 [x1]) = x1 :: (reverse _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            :as (gen_1 : sko_ty_0 list))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                :scheme (not (gen_1 <> []) ==> φ gen_1 x1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        && (gen_1 <> [] && φ (List.tl gen_1) x1 ==> φ gen_1 x1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Split (let (_x_0 : bool) = gen_1 <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_1 : sko_ty_0 list) = [x1] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_2 : bool)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           = reverse (append gen_1 _x_1) = x1 :: (reverse gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_3 : sko_ty_0 list) = List.tl gen_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (_x_0 || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       && (not (_x_0 && (reverse (append _x_3 _x_1) = x1 :: (reverse _x_3)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       :cases [gen_1 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || (reverse (append gen_1 [x1]) = x1 :: (reverse gen_1));
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_0 : sko_ty_0 list) = List.tl gen_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_1 : sko_ty_0 list) = [x1] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               not (gen_1 <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || not (reverse (append _x_0 _x_1) = x1 :: (reverse _x_0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || (reverse (append gen_1 _x_1) = x1 :: (reverse gen_1))])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              x <> [] || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.449s, "1.2"] x <> [] || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                [reverse, reverse]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Success!

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Imandra proves this fact automatically, and its inductive proof actually involves a nested subinduction.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Imandra has powerful techniques for automating complex inductions, involving simplification, destructor elimination, generalization and more. But it's often the case that nested inductions actually suggest lemmas that could be of general use.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            To illustrate this, let's take a look at one of the interesting subgoals in our proof:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse (append gen_1 (x1 :: [])) = x1 :: (reverse gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            This looks like a very nice fact. Instead of Imandra having to derive this fact as a subinduction, let's prove it as a theorem itself and install it as a rewrite rule. Then we will be able to apply it later and shorten our subsequent proofs.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            In [20]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            theorem rev_app_single x y = reverse (append x [y]) = y :: (reverse x) [@@induct] [@@rewrite]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Out[20]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            val rev_app_single : 'a list -> 'a -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse (append x [y]) = y :: (reverse x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             reverse (append x [y]) = y :: (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            The recursive terms in the conjecture suggest 2 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Subsumption and merging reduces this to 1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (not (x <> []) ==> φ x y) && (x <> [] && φ (List.tl x) y ==> φ x y).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             C1. reverse (append x [y]) = y :: (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            But simplification reduces this to true, using the definitions of append and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             H1. reverse (append (List.tl x) [y]) = y :: (reverse (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             reverse (append x [y]) = y :: (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            This simplifies, using the definitions of append and reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Subgoal 1.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             H1. reverse (append (List.tl x) [y]) = y :: (reverse (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             append (reverse (append (List.tl x) [y])) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = y :: (append (reverse (List.tl x)) [List.hd x])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            But we verify Subgoal 1.1' by recursive unrolling.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            ground_instances:5
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            definitions:8
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            0.203s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            details:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            smt_stats:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            num checks:11
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            arith-make-feasible:8
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            arith-max-columns:4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rlimit count:32176
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            mk clause:120
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            datatype occurs check:119
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            mk bool var:222
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            datatype splits:26
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            decisions:74
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            propagations:97
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            conflicts:11
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            datatype accessor ax:24
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            minimized lits:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            datatype constructor ax:38
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            num allocs:3381610421
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            final checks:14
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            added eqs:283
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            del clause:93
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            memory:23.080000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            max memory:23.080000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • start[0.203s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse (append ( :var_0: ) [( :var_1: )])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                = ( :var_1: ) :: (reverse ( :var_0: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              reverse (append x [y]) = y :: (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.203s, "1"] reverse (append x [y]) = y :: (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                :scheme (not (x <> []) ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        && (x <> [] && φ (List.tl x) y ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_1 : sko_ty_0 list) = [y] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_2 : bool) = reverse (append x _x_1) = y :: (reverse x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_3 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (_x_0 || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       && (not (_x_0 && (reverse (append _x_3 _x_1) = y :: (reverse _x_3)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       :cases [x <> [] || (reverse (append x [y]) = y :: (reverse x));
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_1 : sko_ty_0 list) = [y] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || not (reverse (append _x_0 _x_1) = y :: (reverse _x_0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || (reverse (append x _x_1) = y :: (reverse x))])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : sko_ty_0 list) = List.tl x in let (_x_1 : sko_ty_0 list) = [y] in not (x <> []) || not (reverse (append _x_0 _x_1) = y :: (reverse _x_0)) || (reverse (append x _x_1) = y :: (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[0.165s, "1.1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_1 : sko_ty_0 list) = [y] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      not (x <> []) || not (reverse (append _x_0 _x_1) = y :: (reverse _x_0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      || (reverse (append x _x_1) = y :: (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    let (_x_1 : sko_ty_0 list) = reverse (append _x_0 [y]) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    let (_x_2 : sko_ty_0 list) = reverse _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    let (_x_3 : sko_ty_0 list) = [List.hd x] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    not (x <> []) || not (_x_1 = y :: _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    || (append _x_1 _x_3 = y :: (append _x_2 _x_3))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    [reverse, reverse, append]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : sko_ty_0 list) = List.tl ( :var_0: ) in let (_x_1 : sko_ty_0 list) = reverse (append _x_0 [( :var_1: )]) in let (_x_2 : sko_ty_0 list) = reverse _x_0 in let (_x_3 : sko_ty_0 list) = [List.hd ( :var_0: )] in not (( :var_0: ) <> []) || not (_x_1 = ( :var_1: ) :: _x_2) || (append _x_1 _x_3 = ( :var_1: ) :: (append _x_2 _x_3))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Start (let (_x_0 : sko_ty_0 list) = List.tl ( :var_0: ) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_1 : sko_ty_0 list) = reverse (append _x_0 [( :var_1: )]) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_2 : sko_ty_0 list) = reverse _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_3 : sko_ty_0 list) = [List.hd ( :var_0: )] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               not (( :var_0: ) <> []) || not (_x_1 = ( :var_1: ) :: _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || (append _x_1 _x_3 = ( :var_1: ) :: (append _x_2 _x_3)) :time 0.014s)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      x <> [] || (reverse (append x [y]) = y :: (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.165s, "1.2"] x <> [] || (reverse (append x [y]) = y :: (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [reverse, reverse, append, reverse, append]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Now, if we were to try to prove our original reverse (reverse x) goal again, we'd get a shorter proof that makes use of our new lemma:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    In [21]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    verify (fun x -> reverse (reverse x) = x) [@@induct]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Out[21]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    - : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    reverse (reverse x) = x.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    We shall induct according to a scheme derived from reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (not (x <> []) ==> φ x) && (x <> [] && φ (List.tl x) ==> φ x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. reverse (reverse (List.tl x)) = List.tl x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definition of reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. reverse (reverse (List.tl x)) = List.tl x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     reverse (append (reverse (List.tl x)) [List.hd x]) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the rewrite rule
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    rev_app_single.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:rw rev_app_single)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:induct reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    definitions:3
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    0.132s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • start[0.132s, "Goal"] reverse (reverse ( :var_0: )) = ( :var_0: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.132s, "1"] reverse (reverse x) = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        :scheme (not (x <> []) ==> φ x) && (x <> [] && φ (List.tl x) ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_1 : bool) = reverse (reverse x) = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_2 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (_x_0 || _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               && (not (_x_0 && (reverse (reverse _x_2) = _x_2)) || _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               :cases [x <> [] || (reverse (reverse x) = x);
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       not (x <> []) || not (reverse (reverse _x_0) = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || (reverse (reverse x) = x)])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : sko_ty_0 list) = List.tl x in not (x <> []) || not (reverse (reverse _x_0) = _x_0) || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.096s, "1.1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              not (x <> []) || not (reverse (reverse _x_0) = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_1 : sko_ty_0 list) = reverse _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            not (x <> []) || not (reverse _x_1 = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            || (reverse (append _x_1 [List.hd x]) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rev_app_single
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                x <> [] || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • start[0.096s, "1.2"] x <> [] || (reverse (reverse x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  [reverse, reverse]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Excellent!

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Reverse Append

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Now, let's try to prove another interesting conjecture about our functions append and reverse:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              reverse (append x y) = append (reverse y) (reverse x)

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Actually, first let's pretend we made a mistake in formulating this conjecture, and accidentally swapped the x and y on the RHS (Right Hand Side) of the equality. Let's constrain the types of lists involved to make the counterexample especially easy to read:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              In [22]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              verify (fun (x : int list) y -> reverse (append x y) = append (reverse x) (reverse y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Out[22]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              - : Z.t list -> Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              module CX : sig val x : Z.t list val y : Z.t list end
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Counterexample (after 21 steps, 0.025s):
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let x : int list = [0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let y : int list = [2]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Refuted
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              proof attempt
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ground_instances:21
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              definitions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              inductions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              0.025s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              details:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              smt_stats:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              num checks:43
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              rlimit count:22743
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              mk clause:110
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype occurs check:1009
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              mk bool var:1357
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype splits:242
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              decisions:902
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              propagations:809
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              conflicts:62
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype accessor ax:213
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              minimized lits:2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype constructor ax:467
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              final checks:68
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              added eqs:2938
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              del clause:18
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              memory:23.480000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              max memory:23.480000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              num allocs:4398111145.000000
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.025s]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  reverse (append ( :var_0: ) ( :var_1: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  = append (reverse ( :var_0: )) (reverse ( :var_1: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2161/server x_1304/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2161/server y_1305/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2161/server y_1305/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (reverse_2161/server x_1304/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (reverse_2161/server (append_2156/server x_1304/client y_1305/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (append_2156/server x_1304/client y_1305/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_2155/server| (reverse_2161/server x_1304/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2161/s…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (reverse_2161/server (|get.::.1_2155/server| y_1305/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|::| (|get.::.…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (reverse_2161/server (|get.::.1_2155/server| y_1305/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2161/server (|get.::.1_2155/server| x_1304/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|::| (|get.::.…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2161/server (|get.::.1_2155/server| x_1304/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_2155/server| (append_2156/server x_1304/cl…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_2155/server| (append_2156/server x_1304/client y_1305/client)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (append_2156/server (|get.::.1_2155/server| x_1304/client) y_1305/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_2155/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (reverse_2161/server (|get.::.1_2155/server| x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_2155/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (|get.::.1_2155/server| (reverse_2161/server x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_2155/server| (|get.::.1_2155/server| x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2155/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2155/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (|get.::.1_2155/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_2155/server| (append_2156/server x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_2155/server| (|get.::.1_2155/server| x_1304/client)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (append_2156/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_2155/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (reverse_2161/server (|get.::.1_2155/server| y_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2155/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (reverse_2161/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2155/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • Sat (Some let x : int list = [(Z.of_nativeint (0n))] let y : int list = [(Z.of_nativeint (2n))] )

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Ah! Let's do it right this time:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        In [23]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        verify (fun (x : int list) y -> reverse (append x y) = append (reverse y) (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Out[23]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        - : Z.t list -> Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Unknown (Verified up to bound 100)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expanded:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (append x y)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (append x y))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (reverse (List.tl x))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (append x y)))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (reverse y)))) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl y) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (reverse (List.tl y)) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (append x y)) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl y in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (append x y))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (append x y))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl y)))))) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (append x y)) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (reverse (List.tl y))))) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl x) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append x y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl y))))))) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl y)) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (reverse y))))) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl x)))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse y))))))) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl x))))) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (reverse (List.tl x)) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl x)) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl x))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl x)) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (append x y)) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl y in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl x)))) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (reverse (List.tl y))) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (reverse y))) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x)))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (reverse (List.tl x)))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (reverse (List.tl y)))) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (append x y)))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl x))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl x in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (append x y)) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (append x y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (reverse y)) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl x) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (reverse (List.tl x))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl x))) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl x) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl y)) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (append x y)))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse y)))))) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl y) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl y))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl y in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl y) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (append x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (append x y)) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (append x y))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl x) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl x in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl y in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (append x y))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl x in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl x in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl x)) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (append x y)))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        blocked:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl x in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl x)) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl x))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl y)))))))) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (append x y))) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl x))))))) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = append x y in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl x)))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl y)) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (append x y))))) in append (List.tl (List.tl (reverse (List.tl _x_0)))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl x) in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (append x y) in append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (append x y)))) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (append x y)) in append (List.tl (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))) y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • append (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (reverse y)))))))) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • reverse (List.tl (List.tl (List.tl (List.tl (List.tl y)))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl y) in append (List.tl (List.tl (List.tl (reverse (List.tl _x_0))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl y))) in append (reverse (List.tl _x_0)) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl y in append (List.tl (List.tl (List.tl (List.tl (reverse (List.tl _x_0)))))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (append x y)))))) in append (List.tl (reverse (List.tl _x_0))) [List.hd _x_0]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        proof attempt
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        ground_instances:100
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        definitions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        inductions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        0.251s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • start[0.251s]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            reverse (append ( :var_0: ) ( :var_1: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            = append (reverse ( :var_1: )) (reverse ( :var_0: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server y_1308/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server x_1307/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server x_1307/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (reverse_2225/server y_1308/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_2225/server (append_2220/server x_1307/client y_1308/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (append_2220/server x_1307/client y_1308/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_2219/server| (reverse_2225/server y_1308/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (reverse_2225/s…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (reverse_2225/server (|get.::.1_2219/server| x_1307/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (|::| (|get.::.…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (reverse_2225/server (|get.::.1_2219/server| x_1307/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server (|get.::.1_2219/server| y_1308/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|::| (|get.::.…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server (|get.::.1_2219/server| y_1308/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_2219/server| (append_2220/server x_1307/cl…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_2219/server| (append_2220/server x_1307/client y_1308/client)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (append_2220/server (|get.::.1_2219/server| x_1307/client) y_1308/client)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (reverse_2225/server (|get.::.1_2219/server| x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (|get.::.1_2219/server| (reverse_2225/server y_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_2219/server| (|get.::.1_2219/server| x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_2219/server| (|get.::.1_2219/server| x_1307/client)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (reverse_2225/server (|get.::.1_2219/server| y_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_2219/server| (append_2220/server x_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|get.::.1_2219/server| (|get.::.1_2219/server| x_1307/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      y_1308/clien…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_2219/server| (|get.::.1_2219/server| y_130…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_2219/server| (|get.::.1_2219/server| y_1308/client)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_2219/server| (|get.::.1_2219/server| x…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (append_2220/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (|get.::.1_2219/server| (|get.::.1_2219/server| x_…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (|get.::.1_2219/server| (|get.::.1_2219/server| y…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (reverse_2225…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_22…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (reverse_2225/server
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_2219/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_2219/…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Our conjecture has passed through to depth 100 recursive unrolling, so we feel pretty confident it is true. Let's try to prove it by induction:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In [24]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                verify (fun x y -> reverse (append x y) = append (reverse y) (reverse x)) [@@induct]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Out[24]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                - : 'a list -> 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse (append x y) = append (reverse y) (reverse x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                The recursive terms in the conjecture suggest 3 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subsumption and merging reduces this to 2.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Only 1 of those schemes are unflawed.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (x <> []) ==> φ x y) && (x <> [] && φ (List.tl x) y ==> φ x y).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definitions of append and reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse y = append (reverse y) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (y <> []) ==> φ x y) && (y <> [] && φ x (List.tl y) ==> φ x y).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C2. reverse y = append (reverse y) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definitions of append and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> [] || (reverse (List.tl y) = append (reverse (List.tl y)) [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse y = append (reverse y) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definition of reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. reverse (List.tl y) = append (reverse (List.tl y)) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (reverse (List.tl y)) [List.hd y]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse (List.tl y)) [List.hd y]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We can eliminate destructors by the following substitution:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 y -> y1 :: y2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1'':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. reverse y2 = append (reverse y2) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (reverse y2) [y1] = append (append (reverse y2) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Cross-fertilizing with:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 reverse y2 = append (reverse y2) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1''':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (reverse y2) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse y2) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (y2 <> []) ==> φ x y1 y2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 && (y2 <> [] && φ x y1 (List.tl y2) ==> φ x y1 y2).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1'''.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. y2 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C2. append (append (reverse y2) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse y2) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definitions of append and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1'''.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. y2 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     || (append (append (reverse (List.tl y2)) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = append (append (reverse (List.tl y2)) [y1]) [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (reverse y2) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse y2) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definition of reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1'''.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. y2 <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. append (append (reverse (List.tl y2)) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse (List.tl y2)) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (append (reverse (List.tl y2)) [List.hd y2]) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (append (reverse (List.tl y2)) [List.hd y2]) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We can eliminate destructors by the following substitution:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 y2 -> y21 :: y22
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1'''.1'':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. append (append (reverse y22) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse y22) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (append (reverse y22) [y21]) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (append (reverse y22) [y21]) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Candidates for generalization:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 reverse y22
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2'.1'''.1''':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. append (append gen_1 []) [y1] = append (append gen_1 [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (append gen_1 [y21]) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (append gen_1 [y21]) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Aborting proof attempt for _verify_target.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:induct reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Checkpoints:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. append (append gen_1 []) [y1] = append (append gen_1 [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (append gen_1 [y21]) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (append gen_1 [y21]) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append (append (reverse y2) []) [y1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (append (reverse y2) [y1]) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse y = append (reverse y) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Error:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Maximum induction depth reached (3). You can set this with #max_induct.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  At <none>:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Success! And in a proof with three inductions! If we inspect the proof, we'll see that those subsequent inductions actually suggest some very useful lemmas.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In fact, this phenomenon of goals proved by subinductions suggesting useful lemmas happens so often, that we commonly like to work with a very low #max_induct value. Let's attempt the same proof as above, but instruct Imandra to only perform inductions of depth 1 (i.e., no subinductions allowed):

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In [25]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                #max_induct 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Out[25]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In [26]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                verify (fun x y -> reverse (append x y) = append (reverse y) (reverse x)) [@@induct]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Out[26]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                - : 'a list -> 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse (append x y) = append (reverse y) (reverse x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                The recursive terms in the conjecture suggest 3 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subsumption and merging reduces this to 2.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Only 1 of those schemes are unflawed.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (x <> []) ==> φ x y) && (x <> [] && φ (List.tl x) y ==> φ x y).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definitions of append and reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse y = append (reverse y) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Aborting proof attempt for _verify_target.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Checkpoints:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse y = append (reverse y) []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Error:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Maximum induction depth reached (1). You can set this with #max_induct.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  At <none>:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We see above that Imandra is unable to prove our theorem with #max_induct set to 1. However, by inspecting the subgoal Imandra was working on before it hit its #max_induct limit (nicely presented to us as a Checkpoint at the end of the proof attempt), we can find a useful lemma we should prove!

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In particular, note this checkpoint:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                gen_1 = append gen_1 []

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This, oriented as append gen_1 [] = gen_1 is a great rule. Note that it really does require induction, as append recurses on its first argument, not its second.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Let's ask Imandra to prove this as a theorem and to install it as a rewrite rule:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In [27]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                theorem append_nil x = append x [] = x [@@induct] [@@rewrite]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Out[27]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                val append_nil : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                append x [] = x.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 append x [] = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (x <> []) ==> φ x) && (x <> [] && φ (List.tl x) ==> φ x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. append x [] = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definition of append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. append (List.tl x) [] = List.tl x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 append x [] = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definition of append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                definitions:2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                0.076s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • start[0.076s, "Goal"] append ( :var_0: ) [] = ( :var_0: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  append x [] = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[0.076s, "1"] append x [] = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    :scheme (not (x <> []) ==> φ x) && (x <> [] && φ (List.tl x) ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : bool) = append x [] = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_2 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (_x_0 || _x_1) && (not (_x_0 && (append _x_2 [] = _x_2)) || _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           :cases [x <> [] || (append x [] = x);
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   not (x <> []) || not (append _x_0 [] = _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || (append x [] = x)])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : sko_ty_0 list) = List.tl x in not (x <> []) || not (append _x_0 [] = _x_0) || (append x [] = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.040s, "1.1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          not (x <> []) || not (append _x_0 [] = _x_0) || (append x [] = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        append
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          x <> [] || (append x [] = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.040s, "1.2"] x <> [] || (append x [] = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            append
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Success! Now, let's return to our main goal and see if we progress any further (still with #max_induct 1) now that we have append_nil available as a rewrite rule:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        In [28]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        verify (fun x y -> reverse (append x y) = append (reverse y) (reverse x)) [@@induct]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Out[28]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        - : 'a list -> 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        reverse (append x y) = append (reverse y) (reverse x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        The recursive terms in the conjecture suggest 3 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subsumption and merging reduces this to 2.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Only 1 of those schemes are unflawed.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (not (x <> []) ==> φ x y) && (x <> [] && φ (List.tl x) y ==> φ x y).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         C1. reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definitions of append and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        reverse, and the rewrite rule append_nil.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H1. reverse (append (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = append (reverse y) (reverse (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        This simplifies, using the definitions of append and reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H1. reverse (append (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = append (reverse y) (reverse (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append (reverse (append (List.tl x) y)) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = append (reverse y) (append (reverse (List.tl x)) [List.hd x])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        We can eliminate destructors by the following substitution:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         x -> x1 :: x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.1'':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. reverse (append x2 y) = append (reverse y) (reverse x2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append (reverse (append x2 y)) [x1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = append (reverse y) (append (reverse x2) [x1])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Cross-fertilizing with:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         reverse (append x2 y) = append (reverse y) (reverse x2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.1''':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append (append (reverse y) (reverse x2)) [x1]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = append (reverse y) (append (reverse x2) [x1])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Candidates for generalization:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         reverse x2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         reverse y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.1'''':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append (append gen_2 gen_1) [x1] = append gen_2 (append gen_1 [x1])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Aborting proof attempt for _verify_target.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:rw append_nil)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Checkpoints:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append (append gen_2 gen_1) [x1] = append gen_2 (append gen_1 [x1])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Error:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Maximum induction depth reached (1). You can set this with #max_induct.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          At <none>:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Yes! If we inspect Imandra's aborted proof, we see that our rewrite rule append_nil was applied exactly where we wanted it, and we now see another interesting subgoal which looks quite familiar:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        append (append gen_1 gen_2) (x1 :: []) = append gen_1 (append gen_2 (x1 :: []))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        That is, Imandra has derived, as a subgoal to be proved, an instance of the associativity of append!

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        If we set our #max_induct to 2, then Imandra would finish this proof automatically.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But, this fact about append seems very useful and of general purpose, so let's prove it as a rewrite rule and make it available to our subsequent proof efforts.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        In [29]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        theorem assoc_append x y z = append x (append y z) = append (append x y) z [@@induct] [@@rewrite]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Out[29]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        val assoc_append : 'a list -> 'a list -> 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        append x (append y z) = append (append x y) z.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        The recursive terms in the conjecture suggest 3 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subsumption and merging reduces this to 2.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Only 1 of those schemes are unflawed.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (not (x <> []) ==> φ x y z) && (x <> [] && φ (List.tl x) y z ==> φ x y z).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         C1. append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definition of append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H1. append (List.tl x) (append y z) = append (append (List.tl x) y) z
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definition of append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        definitions:5
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        0.086s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • start[0.086s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            append ( :var_0: ) (append ( :var_1: ) ( :var_2: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            = append (append ( :var_0: ) ( :var_1: )) ( :var_2: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.086s, "1"] append x (append y z) = append (append x y) z
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            :scheme (not (x <> []) ==> φ x y z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    && (x <> [] && φ (List.tl x) y z ==> φ x y z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_1 : sko_ty_0 list) = append y z in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_2 : bool) = append x _x_1 = append (append x y) z in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_3 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (_x_0 || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   && (not (_x_0 && (append _x_3 _x_1 = append (append _x_3 y) z))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   :cases [x <> [] || (append x (append y z) = append (append x y) z);
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : sko_ty_0 list) = append y z in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || not (append _x_0 _x_1 = append (append _x_0 y) z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || (append x _x_1 = append (append x y) z)])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_0 : sko_ty_0 list) = List.tl x in let (_x_1 : sko_ty_0 list) = append y z in not (x <> []) || not (append _x_0 _x_1 = append (append _x_0 y) z) || (append x _x_1 = append (append x y) z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.042s, "1.1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_1 : sko_ty_0 list) = append y z in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  not (x <> []) || not (append _x_0 _x_1 = append (append _x_0 y) z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  || (append x _x_1 = append (append x y) z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                [append, append, append]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  x <> [] || (append x (append y z) = append (append x y) z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[0.042s, "1.2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      x <> [] || (append x (append y z) = append (append x y) z)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    [append, append]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Success! Now, let's go back to our original goal and see if we can prove it with the help of our proved rules (still with max_induct 1):

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In [30]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                verify (fun x y -> reverse (append x y) = append (reverse y) (reverse x)) [@@induct]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Out[30]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                - : 'a list -> 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse (append x y) = append (reverse y) (reverse x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                The recursive terms in the conjecture suggest 3 inductions.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subsumption and merging reduces this to 2.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Only 1 of those schemes are unflawed.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (x <> []) ==> φ x y) && (x <> [] && φ (List.tl x) y ==> φ x y).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definitions of append and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                reverse, and the rewrite rule append_nil.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. reverse (append (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (reverse y) (reverse (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definitions of append and reverse to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. reverse (append (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = append (reverse y) (reverse (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 append (reverse (append (List.tl x) y)) [List.hd x]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 = append (reverse y) (append (reverse (List.tl x)) [List.hd x])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the rewrite rule assoc_append.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def reverse)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:rw append_nil)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:rw assoc_append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:induct append)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                definitions:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                0.149s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • start[0.149s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    reverse (append ( :var_0: ) ( :var_1: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    = append (reverse ( :var_1: )) (reverse ( :var_0: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[0.149s, "1"] reverse (append x y) = append (reverse y) (reverse x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • induction on (functional ?)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    :scheme (not (x <> []) ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            && (x <> [] && φ (List.tl x) y ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • Split (let (_x_0 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : sko_ty_0 list) = reverse y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_2 : bool) = reverse (append x y) = append _x_1 (reverse x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_3 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (_x_0 || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           && (not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (_x_0 && (reverse (append _x_3 y) = append _x_1 (reverse _x_3)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           :cases [x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || (reverse (append x y) = append (reverse y) (reverse x));
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_1 : sko_ty_0 list) = reverse y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || not (reverse (append _x_0 y) = append _x_1 (reverse _x_0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || (reverse (append x y) = append _x_1 (reverse x))])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : sko_ty_0 list) = List.tl x in let (_x_1 : sko_ty_0 list) = reverse y in not (x <> []) || not (reverse (append _x_0 y) = append _x_1 (reverse _x_0)) || (reverse (append x y) = append _x_1 (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.110s, "1.1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_1 : sko_ty_0 list) = reverse y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          not (x <> []) || not (reverse (append _x_0 y) = append _x_1 (reverse _x_0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          || (reverse (append x y) = append _x_1 (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_1 : sko_ty_0 list) = reverse (append _x_0 y) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_2 : sko_ty_0 list) = reverse y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_3 : sko_ty_0 list) = reverse _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_4 : sko_ty_0 list) = [List.hd x] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        not (x <> []) || not (_x_1 = append _x_2 _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        || (append _x_1 _x_4 = append _x_2 (append _x_3 _x_4))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [reverse, reverse, append]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            assoc_append
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            x <> [] || (reverse (append x y) = append (reverse y) (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • start[0.110s, "1.2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                x <> [] || (reverse (append x y) = append (reverse y) (reverse x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              [append, reverse, append]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              append_nil
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Success! And our rewrite rules were used exactly where we'd hoped.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Happy proving!