Verifying Merge Sort in Imandra

Merge sort is a widely used efficient general purpose sorting algorithm invented by John von Neumann in 1945. It is a prototypical divide and conquer algorithm and forms the basis of standard library sorting functions in languages like OCaml, Java and Python.

Let's verify it in Imandra!

The main idea

The main idea of merge sort is as follows:

  • Divide the input list into sublists
  • Sort the sublists
  • Merge the sorted sublists to obtain a sorted version of the original list

Merging two lists

We'll start by defining a function to merge two lists in a "sorted" way. Note that our termination proof involves both arguments l and m, using a lexicographic product (via the [@@adm l,m] admission annotation).

In [1]:
let rec merge (l : int list) (m : int list) =
 match l, m with
  | [], _ -> m
  | _, [] -> l
  | a :: ls, b :: ms ->
    if a < b then
     a :: (merge ls m)
    else
     b :: (merge l ms)
[@@adm l, m]
Out[1]:
val merge : Z.t list -> Z.t list -> Z.t list = <fun>
termination proof

Termination proof

call `merge (List.tl l) m` from `merge l m`
original:merge l m
sub:merge (List.tl l) m
original ordinal:Ordinal.of_list [Ordinal.Int (_cnt l); Ordinal.Int (_cnt m)]
sub ordinal:Ordinal.of_list [Ordinal.Int (_cnt (List.tl l)); Ordinal.Int (_cnt m)]
path:[List.hd l < List.hd m && m <> [] && l <> []]
proof:
detailed proof
ground_instances:8
definitions:0
inductions:0
search_time:
0.014s
details:
Expand
smt_stats:
arith offset eqs:6
num checks:18
arith assert lower:134
arith tableau max rows:20
arith tableau max columns:42
arith pivots:99
rlimit count:49758
mk clause:206
datatype occurs check:224
mk bool var:638
arith assert upper:131
datatype splits:40
decisions:304
arith row summations:223
arith bound prop:3
propagations:336
interface eqs:10
conflicts:36
arith fixed eqs:85
arith assume eqs:10
datatype accessor ax:84
minimized lits:2
arith conflicts:6
arith num rows:20
arith assert diseq:30
datatype constructor ax:94
num allocs:12018881
final checks:29
added eqs:867
del clause:149
arith eq adapter:106
memory:15.760000
max memory:15.980000
Expand
  • start[0.014s]
      let (_x_0 : int) = List.hd m in
      let (_x_1 : bool) = m <> [] in
      let (_x_2 : bool) = count.list mk_nat m >= 0 in
      let (_x_3 : int list) = List.tl l in
      let (_x_4 : bool) = List.hd _x_3 < _x_0 in
      let (_x_5 : bool) = _x_1 && _x_3 <> [] in
      let (_x_6 : Ordinal.t) = if … <> [] then … else … in
      (List.hd l < _x_0)
      && (_x_1
          && (l <> []
              && ((count.list mk_nat l >= 0)
                  && (_x_2 && ((count.list mk_nat _x_3 >= 0) && _x_2)))))
      ==> (not (_x_4 && _x_5) && not (not _x_4 && _x_5))
          || Ordinal.( << ) _x_6 _x_6
  • simplify
    into:
    let (_x_0 : int list) = List.tl l in
    let (_x_1 : int) = count.list mk_nat _x_0 in
    let (_x_2 : Ordinal.t) = Ordinal.Int 1 in
    let (_x_3 : int) = count.list mk_nat m in
    let (_x_4 : Ordinal.t) = Ordinal.Int _x_3 in
    let (_x_5 : int) = count.list mk_nat l in
    let (_x_6 : int) = List.hd m in
    let (_x_7 : bool) = m <> [] in
    let (_x_8 : bool) = _x_6 <= List.hd _x_0 in
    let (_x_9 : bool) = _x_0 <> [] in
    Ordinal.( << ) (Ordinal.plus (Ordinal.shift (Ordinal.Int _x_1) _x_2) _x_4)
    (Ordinal.plus (Ordinal.shift (Ordinal.Int _x_5) _x_2) _x_4)
    || not
       (not (_x_6 <= List.hd l) && _x_7 && l <> [] && (_x_5 >= 0) && (_x_3 >= 0)
        && (_x_1 >= 0))
    || (not (not _x_8 && _x_7 && _x_9) && not (_x_8 && _x_7 && _x_9))
    expansions:
    []
    rewrite_steps:
      forward_chaining:
      • unroll
        expr:
        (let ((a!1 (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                      (|count.list_384/se…
        expansions:
        • unroll
          expr:
          (let ((a!1 (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                        (|count.list_384/se…
          expansions:
          • unroll
            expr:
            (|Ordinal.shift| (|Ordinal.Int_79/boot|
                               (|count.list_384/server|
                            …
            expansions:
            • unroll
              expr:
              (|count.list_384/server| (|get.::.1_368/server| l_369/server))
              expansions:
              • unroll
                expr:
                (|Ordinal.plus| (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                   (|count.l…
                expansions:
                • unroll
                  expr:
                  (|Ordinal.shift| (|Ordinal.Int_79/boot| (|count.list_384/server| l_369/server))
                                   (|O…
                  expansions:
                  • unroll
                    expr:
                    (|count.list_384/server| m_370/server)
                    expansions:
                    • unroll
                      expr:
                      (|count.list_384/server| l_369/server)
                      expansions:
                      • Unsat
                      call `merge l (List.tl m)` from `merge l m`
                      original:merge l m
                      sub:merge l (List.tl m)
                      original ordinal:Ordinal.of_list [Ordinal.Int (_cnt l); Ordinal.Int (_cnt m)]
                      sub ordinal:Ordinal.of_list [Ordinal.Int (_cnt l); Ordinal.Int (_cnt (List.tl m))]
                      path:[not (List.hd l < List.hd m) && m <> [] && l <> []]
                      proof:
                      detailed proof
                      ground_instances:12
                      definitions:0
                      inductions:0
                      search_time:
                      0.023s
                      details:
                      Expand
                      smt_stats:
                      arith offset eqs:20
                      num checks:26
                      arith assert lower:458
                      arith tableau max rows:36
                      arith tableau max columns:70
                      arith pivots:208
                      rlimit count:33926
                      mk clause:428
                      datatype occurs check:490
                      mk bool var:1276
                      arith assert upper:353
                      datatype splits:73
                      decisions:644
                      arith row summations:501
                      arith bound prop:24
                      propagations:1261
                      interface eqs:2
                      conflicts:67
                      arith fixed eqs:213
                      arith assume eqs:2
                      datatype accessor ax:148
                      minimized lits:6
                      arith conflicts:12
                      arith num rows:36
                      arith assert diseq:88
                      datatype constructor ax:196
                      num allocs:6019520
                      final checks:35
                      added eqs:1829
                      del clause:333
                      arith eq adapter:287
                      memory:15.890000
                      max memory:15.980000
                      Expand
                      • start[0.023s]
                          let (_x_0 : int) = List.hd l in
                          let (_x_1 : bool) = l <> [] in
                          let (_x_2 : bool) = count.list mk_nat l >= 0 in
                          let (_x_3 : int list) = List.tl m in
                          let (_x_4 : bool) = _x_0 < List.hd _x_3 in
                          let (_x_5 : bool) = _x_3 <> [] && _x_1 in
                          let (_x_6 : Ordinal.t) = if … <> [] then … else … in
                          not (_x_0 < List.hd m)
                          && (m <> []
                              && (_x_1
                                  && (_x_2
                                      && ((count.list mk_nat m >= 0)
                                          && (_x_2 && (count.list mk_nat _x_3 >= 0))))))
                          ==> (not (_x_4 && _x_5) && not (not _x_4 && _x_5))
                              || Ordinal.( << ) _x_6 _x_6
                      • simplify
                        into:
                        let (_x_0 : int list) = List.tl m in
                        let (_x_1 : int) = List.hd l in
                        let (_x_2 : bool) = List.hd _x_0 <= _x_1 in
                        let (_x_3 : bool) = _x_0 <> [] in
                        let (_x_4 : bool) = l <> [] in
                        let (_x_5 : Ordinal.t) = Ordinal.Int 1 in
                        let (_x_6 : int) = count.list mk_nat … in
                        (not (not _x_2 && _x_3 && _x_4) && not (_x_2 && _x_3 && _x_4))
                        || Ordinal.( << )
                           (Ordinal.plus (Ordinal.shift (Ordinal.Int (count.list mk_nat l)) _x_5)
                            (Ordinal.Int (count.list mk_nat _x_0)))
                           (Ordinal.plus (Ordinal.shift (Ordinal.Int …) _x_5) (Ordinal.Int _x_6))
                        || not
                           ((List.hd … <= _x_1) && … <> [] && _x_4 && (… >= 0) && (_x_6 >= 0)
                            && (count.list mk_nat (List.tl …) >= 0))
                        expansions:
                        []
                        rewrite_steps:
                          forward_chaining:
                          • unroll
                            expr:
                            (let ((a!1 (|Ordinal.plus| (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                             …
                            expansions:
                            • unroll
                              expr:
                              (|Ordinal.plus| (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                                 (|count.l…
                              expansions:
                              • unroll
                                expr:
                                (|Ordinal.plus| (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                                   (|count.l…
                                expansions:
                                • unroll
                                  expr:
                                  (|Ordinal.shift| (|Ordinal.Int_79/boot| (|count.list_384/server| l_369/server))
                                                   (|O…
                                  expansions:
                                  • unroll
                                    expr:
                                    (|count.list_384/server| (|get.::.1_368/server| m_370/server))
                                    expansions:
                                    • unroll
                                      expr:
                                      (|count.list_384/server| m_370/server)
                                      expansions:
                                      • unroll
                                        expr:
                                        (|count.list_384/server| l_369/server)
                                        expansions:
                                        • unroll
                                          expr:
                                          (|count.list_384/server|
                                            (|get.::.1_368/server| (|get.::.1_368/server| m_370/server)))
                                          expansions:
                                          • unroll
                                            expr:
                                            (|count.list_384/server| (|get.::.1_368/server| l_369/server))
                                            expansions:
                                            • unroll
                                              expr:
                                              (let ((a!1 (|Ordinal.plus| (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                                               …
                                              expansions:
                                              • unroll
                                                expr:
                                                (let ((a!1 (|get.Ordinal.Cons.2_375/server|
                                                             (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                   …
                                                expansions:
                                                • unroll
                                                  expr:
                                                  (let ((a!1 (|get.Ordinal.Cons.2_375/server|
                                                               (|Ordinal.shift| (|Ordinal.Int_79/boot|
                                                     …
                                                  expansions:
                                                  • Unsat

                                                  Let's experiment a bit with merge.

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

                                                  Observe that merge will merge sorted lists in a manner that respects their sortedness. But, if the lists given to merge aren't sorted, the output of merge need not be sorted.

                                                  Dividing into sublists

                                                  In merge sort, we need to divide a list into sublists. We'll do this in a simple way using the function odds which computes "every other" element of a list. We'll then be able to split a list x into sublists by taking odds x and odds (List.tl x).

                                                  In [5]:
                                                  let rec odds l =
                                                   match l with
                                                    | []  -> []
                                                    | [x] -> [x]
                                                    | x :: y :: rst -> x :: odds rst
                                                  
                                                  Out[5]:
                                                  val odds : 'a list -> 'a list = <fun>
                                                  
                                                  termination proof

                                                  Termination proof

                                                  call `odds (List.tl (List.tl l))` from `odds l`
                                                  original:odds l
                                                  sub:odds (List.tl (List.tl l))
                                                  original ordinal:Ordinal.Int (_cnt l)
                                                  sub ordinal:Ordinal.Int (_cnt (List.tl (List.tl l)))
                                                  path:[(List.tl l) <> [] && l <> []]
                                                  proof:
                                                  detailed proof
                                                  ground_instances:6
                                                  definitions:0
                                                  inductions:0
                                                  search_time:
                                                  0.011s
                                                  details:
                                                  Expand
                                                  smt_stats:
                                                  num checks:14
                                                  arith assert lower:33
                                                  arith tableau max rows:12
                                                  arith tableau max columns:27
                                                  arith pivots:24
                                                  rlimit count:5485
                                                  mk clause:82
                                                  datatype occurs check:29
                                                  mk bool var:196
                                                  arith assert upper:30
                                                  datatype splits:3
                                                  decisions:80
                                                  arith row summations:50
                                                  propagations:87
                                                  conflicts:19
                                                  arith fixed eqs:26
                                                  datatype accessor ax:18
                                                  arith conflicts:2
                                                  arith num rows:12
                                                  arith assert diseq:2
                                                  datatype constructor ax:27
                                                  num allocs:20496089
                                                  final checks:8
                                                  added eqs:160
                                                  del clause:42
                                                  arith eq adapter:30
                                                  memory:15.980000
                                                  max memory:15.980000
                                                  Expand
                                                  • start[0.011s]
                                                      let (_x_0 : ty_0 list) = List.tl l in
                                                      let (_x_1 : int) = count.list (const 0) l in
                                                      let (_x_2 : ty_0 list) = List.tl _x_0 in
                                                      let (_x_3 : int) = count.list (const 0) _x_2 in
                                                      _x_0 <> [] && (l <> [] && ((_x_1 >= 0) && (_x_3 >= 0)))
                                                      ==> not ((List.tl _x_2) <> [] && _x_2 <> [])
                                                          || Ordinal.( << ) (Ordinal.Int _x_3) (Ordinal.Int _x_1)
                                                  • simplify
                                                    into:
                                                    let (_x_0 : ty_0 list) = List.tl l in
                                                    let (_x_1 : ty_0 list) = List.tl _x_0 in
                                                    let (_x_2 : int) = count.list (const 0) _x_1 in
                                                    let (_x_3 : int) = count.list (const 0) l in
                                                    not ((List.tl _x_1) <> [] && _x_1 <> [])
                                                    || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_3)
                                                    || not (_x_0 <> [] && l <> [] && (_x_3 >= 0) && (_x_2 >= 0))
                                                    expansions:
                                                    []
                                                    rewrite_steps:
                                                      forward_chaining:
                                                      • unroll
                                                        expr:
                                                        (let ((a!1 (|Ordinal.Int_79/boot|
                                                                     (|count.list_543/server|
                                                                       (|get.::.1_52…
                                                        expansions:
                                                        • unroll
                                                          expr:
                                                          (|count.list_543/server|
                                                            (|get.::.1_525/server| (|get.::.1_525/server| l_532/server)))
                                                          expansions:
                                                          • unroll
                                                            expr:
                                                            (|count.list_543/server| l_532/server)
                                                            expansions:
                                                            • unroll
                                                              expr:
                                                              (|count.list_543/server|
                                                                (|get.::.1_525/server|
                                                                  (|get.::.1_525/server| (|get.::.1_525/server| …
                                                              expansions:
                                                              • unroll
                                                                expr:
                                                                (let ((a!1 (|Ordinal.Int_79/boot|
                                                                             (|count.list_543/server|
                                                                               (|get.::.1_52…
                                                                expansions:
                                                                • unroll
                                                                  expr:
                                                                  (|count.list_543/server| (|get.::.1_525/server| l_532/server))
                                                                  expansions:
                                                                  • Unsat

                                                                  Let's compute a bit with odds to better understand how it works.

                                                                  In [6]:
                                                                  odds [1;2;3;4;5;6]
                                                                  
                                                                  Out[6]:
                                                                  - : Z.t list = [1;3;5]
                                                                  
                                                                  In [7]:
                                                                  odds (List.tl [1;2;3;4;5;6])
                                                                  
                                                                  Out[7]:
                                                                  - : Z.t list = [2;4;6]
                                                                  

                                                                  Defining Merge Sort

                                                                  We can now put the pieces together and define our main merge_sort function, as follows:

                                                                  In [8]:
                                                                  let rec merge_sort l =
                                                                   match l with
                                                                    | []  -> []
                                                                    | [_] -> l
                                                                    | _ :: ls -> merge (merge_sort (odds l)) (merge_sort (odds ls))
                                                                  
                                                                  Out[8]:
                                                                  val merge_sort : Z.t list -> Z.t list = <fun>
                                                                  Error:
                                                                    unsupported: rejected definition for function merge_sort,
                                                                                 Imandra could not prove termination. 
                                                                                 hint: problematic sub-call from `merge_sort l`
                                                                                 to `merge_sort (odds (List.tl l))`  under path
                                                                                 not Is_a([], List.tl l) and l <> []
                                                                                 could not be proved to decrease (measured subset: (l))
                                                                    See https://docs.imandra.ai/imandra-docs/notebooks/proving-program-termination/
                                                                    At jupyter cell 8:1,0--129
                                                                    1 | let rec merge_sort l =
                                                                    2 |  match l with
                                                                    3 |   | []  -> []
                                                                    4 |   | [_] -> l
                                                                    5 |   | _ :: ls -> merge (merge_sort (odds l)) (merge_sort (odds ls))
                                                                    

                                                                  To admit merge_sort into Imandra, we must prove it terminating. We can do this with a custom measure function which we'll call merge_sort_measure. We'll prove a lemma about odds (by functional induction) to make it clear why the measure is decreasing in every recursive call. We'll install the lemma as a forward_chaining rule, which is a good strategy for making it effective during termination analysis.

                                                                  In [9]:
                                                                  let merge_sort_measure x =
                                                                   Ordinal.of_int (List.length x)
                                                                  
                                                                  Out[9]:
                                                                  val merge_sort_measure : 'a list -> Ordinal.t = <fun>
                                                                  
                                                                  In [10]:
                                                                  theorem odds_len_1 x =
                                                                   x <> [] && List.tl x <> []
                                                                   ==>
                                                                   (List.length (odds x) [@trigger]) < List.length x
                                                                  [@@induct functional odds] [@@forward_chaining]
                                                                  
                                                                  Out[10]:
                                                                  val odds_len_1 : 'a list -> bool = <fun>
                                                                  Goal:
                                                                  
                                                                  (x <> []) && (List.tl x <> []) ==> List.length (odds x) < List.length x.
                                                                  
                                                                  1 nontautological subgoal.
                                                                  
                                                                  We shall induct according to a scheme derived from odds.
                                                                  
                                                                  Induction scheme:
                                                                  
                                                                   (not ((List.tl x) <> [] && x <> []) ==> φ x)
                                                                   && (x <> [] && ((List.tl x) <> [] && φ (List.tl (List.tl x))) ==> φ x).
                                                                  
                                                                  2 nontautological subgoals.
                                                                  
                                                                  Subgoal 2:
                                                                  
                                                                   H0. List.length x <= List.length (odds x)
                                                                   H1. not (x = [])
                                                                   H2. not (List.tl x = [])
                                                                  |---------------------------------------------------------------------------
                                                                   (List.tl x) <> [] && x <> []
                                                                  
                                                                  But simplification reduces this to true, using the forward-chaining rule
                                                                  List.len_nonnegative.
                                                                  
                                                                  Subgoal 1:
                                                                  
                                                                   H0. List.length x <= List.length (odds x)
                                                                   H1. not (x = [])
                                                                   H2. not (List.tl x = [])
                                                                   H3. x <> []
                                                                   H4. (List.tl x) <> []
                                                                   H5. not
                                                                       (List.length (List.tl (List.tl x))
                                                                        <= List.length (odds (List.tl (List.tl x))))
                                                                       || not
                                                                          (not (List.tl (List.tl x) = [])
                                                                           && not (List.tl (List.tl (List.tl x)) = []))
                                                                  |---------------------------------------------------------------------------
                                                                   false
                                                                  
                                                                  This simplifies, using the definitions of List.length and odds to the
                                                                  following 3 subgoals:
                                                                  
                                                                  Subgoal 1.3:
                                                                  
                                                                   H0. x <> []
                                                                   H1. (List.tl x) <> []
                                                                   H2. List.length (List.tl (List.tl x))
                                                                       <= List.length (odds (List.tl (List.tl x)))
                                                                   H3. List.tl (List.tl x) = []
                                                                   H4. List.length (List.tl x) <= List.length (odds (List.tl (List.tl x)))
                                                                  |---------------------------------------------------------------------------
                                                                   false
                                                                  
                                                                  But simplification reduces this to true, using the definitions of List.length
                                                                  and odds.
                                                                  
                                                                  Subgoal 1.2:
                                                                  
                                                                   H0. x <> []
                                                                   H1. (List.tl x) <> []
                                                                   H2. List.length (List.tl (List.tl x))
                                                                       <= List.length (odds (List.tl (List.tl x)))
                                                                   H3. (List.tl (List.tl x)) <> []
                                                                   H4. List.tl (List.tl (List.tl x)) = []
                                                                   H5. List.length (List.tl x) <= List.length (odds (List.tl (List.tl x)))
                                                                  |---------------------------------------------------------------------------
                                                                   false
                                                                  
                                                                  This simplifies, using the definitions of List.length and odds to:
                                                                  
                                                                  Subgoal 1.2':
                                                                  
                                                                   H0. List.tl (List.tl (List.tl x)) = []
                                                                   H1. (List.tl x) <> []
                                                                   H2. x <> []
                                                                   H3. (List.tl (List.tl x)) <> []
                                                                   H4. List.length (List.tl (List.tl (List.tl x))) <= 0
                                                                   H5. List.length (List.tl (List.tl x)) <= 0
                                                                  |---------------------------------------------------------------------------
                                                                   false
                                                                  
                                                                  But simplification reduces this to true, using the definition of List.length.
                                                                  
                                                                  Subgoal 1.1:
                                                                  
                                                                   H0. x <> []
                                                                   H1. (List.tl x) <> []
                                                                   H2. List.length (List.tl x) <= List.length (odds (List.tl (List.tl x)))
                                                                  |---------------------------------------------------------------------------
                                                                   List.length (List.tl (List.tl x))
                                                                   <= List.length (odds (List.tl (List.tl x)))
                                                                  
                                                                  But simplification reduces this to true, using the definition of List.length.
                                                                  
                                                                   Rules:
                                                                      (:def List.length)
                                                                      (:def odds)
                                                                      (:fc List.len_nonnegative)
                                                                      (:induct odds)
                                                                  
                                                                  
                                                                  Proved
                                                                  proof
                                                                  ground_instances:0
                                                                  definitions:22
                                                                  inductions:1
                                                                  search_time:
                                                                  0.235s
                                                                  Expand
                                                                  • start[0.235s, "Goal"]
                                                                      not (( :var_0: ) = []) && not (List.tl ( :var_0: ) = [])
                                                                      ==> List.length (odds ( :var_0: )) < List.length ( :var_0: )
                                                                  • subproof

                                                                    not (List.length x <= List.length (odds x)) || (x = []) || (List.tl x = [])
                                                                    • start[0.235s, "1"]
                                                                        not (List.length x <= List.length (odds x)) || (x = []) || (List.tl x = [])
                                                                    • induction on (functional odds)
                                                                      :scheme (not ((List.tl x) <> [] && x <> []) ==> φ x)
                                                                              && (x <> [] && ((List.tl x) <> [] && φ (List.tl (List.tl x)))
                                                                                  ==> φ x)
                                                                    • Split (let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                             let (_x_1 : bool) = _x_0 <> [] in
                                                                             let (_x_2 : bool) = x <> [] in
                                                                             let (_x_3 : bool) = not (List.length x <= List.length (odds x)) in
                                                                             let (_x_4 : bool) = not (not (x = []) && not (_x_0 = [])) in
                                                                             let (_x_5 : sko_ty_0 list) = List.tl _x_0 in
                                                                             ((_x_1 && _x_2) || _x_3 || _x_4)
                                                                             && (_x_3 || _x_4
                                                                                 || not
                                                                                    (_x_2 && _x_1
                                                                                     && (not (List.length _x_5 <= List.length (odds _x_5))
                                                                                         || not (not (_x_5 = []) && not (List.tl _x_5 = [])))))
                                                                             :cases [let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                     not (List.length x <= List.length (odds x)) || (x = [])
                                                                                     || (_x_0 = []) || (_x_0 <> [] && x <> []);
                                                                                     let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                                     let (_x_1 : sko_ty_0 list) = List.tl _x_0 in
                                                                                     not (List.length x <= List.length (odds x)) || (x = [])
                                                                                     || (_x_0 = []) || not (x <> []) || not (_x_0 <> [])
                                                                                     || not
                                                                                        (not (List.length _x_1 <= List.length (odds _x_1))
                                                                                         || not (not (_x_1 = []) && not (List.tl _x_1 = [])))])
                                                                      • subproof
                                                                        let (_x_0 : sko_ty_0 list) = List.tl x in let (_x_1 : sko_ty_0 list) = List.tl _x_0 in not (List.length x <= List.length (odds x)) || (x = []) || (_x_0 = []) || not (x <> []) || not (_x_0 <> []) || not (not (List.length _x_1 <= List.length (odds _x_1)) || not (not (_x_1 = []) && not (List.tl _x_1 = [])))
                                                                        • start[0.234s, "1"]
                                                                            let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                            let (_x_1 : sko_ty_0 list) = List.tl _x_0 in
                                                                            not (List.length x <= List.length (odds x)) || (x = []) || (_x_0 = [])
                                                                            || not (x <> []) || not (_x_0 <> [])
                                                                            || not
                                                                               (not (List.length _x_1 <= List.length (odds _x_1))
                                                                                || not (not (_x_1 = []) && not (List.tl _x_1 = [])))
                                                                        • simplify
                                                                          into:
                                                                          let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                          let (_x_1 : bool) = (x = []) || (_x_0 = []) in
                                                                          let (_x_2 : sko_ty_0 list) = List.tl _x_0 in
                                                                          let (_x_3 : int) = List.length (odds _x_2) in
                                                                          let (_x_4 : bool) = List.length _x_2 <= _x_3 in
                                                                          let (_x_5 : bool) = _x_1 || not _x_4 in
                                                                          let (_x_6 : bool) = _x_2 = [] in
                                                                          let (_x_7 : bool) = not (List.length _x_0 <= _x_3) in
                                                                          (_x_5 || not _x_6 || _x_7)
                                                                          && (_x_5 || _x_6 || not (List.tl _x_2 = []) || _x_7)
                                                                          && (_x_1 || _x_4 || _x_7)
                                                                          expansions:
                                                                          [List.length, odds, List.length, List.length, odds, List.length, List.length,
                                                                           odds, List.length]
                                                                          rewrite_steps:
                                                                            forward_chaining:
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • List.len_nonnegative
                                                                            • Subproof
                                                                            • Subproof
                                                                            • Subproof
                                                                        • subproof
                                                                          let (_x_0 : sko_ty_0 list) = List.tl x in not (List.length x <= List.length (odds x)) || (x = []) || (_x_0 = []) || (_x_0 <> [] && x <> [])
                                                                          • start[0.234s, "2"]
                                                                              let (_x_0 : sko_ty_0 list) = List.tl x in
                                                                              not (List.length x <= List.length (odds x)) || (x = []) || (_x_0 = [])
                                                                              || (_x_0 <> [] && x <> [])
                                                                          • simplify
                                                                            into:
                                                                            true
                                                                            expansions:
                                                                            []
                                                                            rewrite_steps:
                                                                              forward_chaining:
                                                                              • List.len_nonnegative
                                                                              • List.len_nonnegative

                                                                      Now we're ready to define merge_sort. Imandra is able prove it terminating using our custom measure (and lemma!).

                                                                      In [11]:
                                                                      let rec merge_sort l =
                                                                       match l with
                                                                        | []  -> []
                                                                        | [_] -> l
                                                                        | _ :: ls -> merge (merge_sort (odds l)) (merge_sort (odds ls))
                                                                      [@@measure merge_sort_measure l]
                                                                      
                                                                      Out[11]:
                                                                      val merge_sort : Z.t list -> Z.t list = <fun>
                                                                      
                                                                      termination proof

                                                                      Termination proof

                                                                      call `merge_sort (odds l)` from `merge_sort l`
                                                                      original:merge_sort l
                                                                      sub:merge_sort (odds l)
                                                                      original ordinal:merge_sort_measure l
                                                                      sub ordinal:merge_sort_measure (odds l)
                                                                      path:[not Is_a([], List.tl l) && l <> []]
                                                                      proof:
                                                                      detailed proof
                                                                      ground_instances:2
                                                                      definitions:0
                                                                      inductions:0
                                                                      search_time:
                                                                      0.008s
                                                                      details:
                                                                      Expand
                                                                      smt_stats:
                                                                      num checks:6
                                                                      arith assert lower:6
                                                                      arith tableau max rows:2
                                                                      arith tableau max columns:6
                                                                      arith pivots:2
                                                                      rlimit count:5220
                                                                      mk clause:4
                                                                      datatype occurs check:20
                                                                      mk bool var:43
                                                                      arith assert upper:2
                                                                      datatype splits:4
                                                                      decisions:10
                                                                      arith row summations:1
                                                                      propagations:5
                                                                      conflicts:5
                                                                      datatype accessor ax:7
                                                                      arith num rows:2
                                                                      datatype constructor ax:8
                                                                      num allocs:2971707499
                                                                      final checks:4
                                                                      added eqs:31
                                                                      del clause:3
                                                                      arith eq adapter:1
                                                                      memory:19.970000
                                                                      max memory:27.090000
                                                                      Expand
                                                                      • start[0.008s]
                                                                          let (_x_0 : int) = List.length l in
                                                                          let (_x_1 : int) = if _x_0 >= 0 then _x_0 else 0 in
                                                                          let (_x_2 : int list) = odds l in
                                                                          let (_x_3 : int) = List.length _x_2 in
                                                                          let (_x_4 : int) = if _x_3 >= 0 then _x_3 else 0 in
                                                                          let (_x_5 : bool) = not (not Is_a([], List.tl _x_2) && _x_2 <> []) in
                                                                          not Is_a([], List.tl l) && (l <> [] && ((_x_1 >= 0) && (_x_4 >= 0)))
                                                                          ==> (_x_5 && _x_5) || Ordinal.( << ) (Ordinal.Int _x_4) (Ordinal.Int _x_1)
                                                                      • simplify
                                                                        into:
                                                                        let (_x_0 : int list) = odds … in
                                                                        let (_x_1 : int) = List.length _x_0 in
                                                                        let (_x_2 : int) = List.length … in
                                                                        Ordinal.( << ) (Ordinal.Int (if _x_1 >= 0 then _x_1 else 0))
                                                                        (Ordinal.Int (if _x_2 >= 0 then _x_2 else 0))
                                                                        || not (not Is_a([], List.tl …) && … <> [])
                                                                        || not (not Is_a([], List.tl _x_0) && _x_0 <> [])
                                                                        expansions:
                                                                        []
                                                                        rewrite_steps:
                                                                          forward_chaining:
                                                                          • unroll
                                                                            expr:
                                                                            (|List.length_941/server| (odds_946/server l_952/server))
                                                                            expansions:
                                                                            • unroll
                                                                              expr:
                                                                              (let ((a!1 (ite (>= (|List.length_941/server| (odds_946/server l_952/server)) 0)
                                                                                              (|L…
                                                                              expansions:
                                                                              • Unsat
                                                                              call `merge_sort (odds (List.tl l))` from `merge_sort l`
                                                                              original:merge_sort l
                                                                              sub:merge_sort (odds (List.tl l))
                                                                              original ordinal:merge_sort_measure l
                                                                              sub ordinal:merge_sort_measure (odds (List.tl l))
                                                                              path:[not Is_a([], List.tl l) && l <> []]
                                                                              proof:
                                                                              detailed proof
                                                                              ground_instances:4
                                                                              definitions:0
                                                                              inductions:0
                                                                              search_time:
                                                                              0.011s
                                                                              details:
                                                                              Expand
                                                                              smt_stats:
                                                                              num checks:10
                                                                              arith assert lower:7
                                                                              arith tableau max rows:4
                                                                              arith tableau max columns:10
                                                                              arith pivots:6
                                                                              rlimit count:3204
                                                                              mk clause:12
                                                                              datatype occurs check:67
                                                                              mk bool var:95
                                                                              arith assert upper:7
                                                                              datatype splits:14
                                                                              decisions:26
                                                                              arith row summations:8
                                                                              propagations:20
                                                                              conflicts:12
                                                                              datatype accessor ax:13
                                                                              arith conflicts:1
                                                                              arith num rows:4
                                                                              datatype constructor ax:19
                                                                              num allocs:2872039753
                                                                              final checks:13
                                                                              added eqs:87
                                                                              del clause:9
                                                                              arith eq adapter:2
                                                                              memory:19.980000
                                                                              max memory:27.090000
                                                                              Expand
                                                                              • start[0.011s]
                                                                                  let (_x_0 : int list) = List.tl … in
                                                                                  let (_x_1 : int) = List.length … in
                                                                                  let (_x_2 : int) = if _x_1 >= 0 then _x_1 else 0 in
                                                                                  let (_x_3 : int list) = odds _x_0 in
                                                                                  let (_x_4 : int) = List.length _x_3 in
                                                                                  let (_x_5 : int) = if _x_4 >= 0 then _x_4 else 0 in
                                                                                  let (_x_6 : bool) = not (not Is_a([], List.tl _x_3) && _x_3 <> []) in
                                                                                  not Is_a([], _x_0) && (… <> [] && ((_x_2 >= 0) && (_x_5 >= 0)))
                                                                                  ==> (_x_6 && _x_6) || Ordinal.( << ) (Ordinal.Int _x_5) (Ordinal.Int _x_2)
                                                                              • simplify
                                                                                into:
                                                                                let (_x_0 : int list) = List.tl … in
                                                                                let (_x_1 : int list) = odds _x_0 in
                                                                                let (_x_2 : int) = List.length _x_1 in
                                                                                let (_x_3 : int) = List.length … in
                                                                                not (not Is_a([], List.tl _x_1) && _x_1 <> [])
                                                                                || Ordinal.( << ) (Ordinal.Int (if _x_2 >= 0 then _x_2 else 0))
                                                                                   (Ordinal.Int (if _x_3 >= 0 then _x_3 else 0))
                                                                                || not (not Is_a([], _x_0) && … <> [])
                                                                                expansions:
                                                                                []
                                                                                rewrite_steps:
                                                                                  forward_chaining:
                                                                                  • unroll
                                                                                    expr:
                                                                                    (let ((a!1 (>= (|List.length_941/server|
                                                                                                     (odds_946/server (|get.::.1_934/server| l_…
                                                                                    expansions:
                                                                                    • unroll
                                                                                      expr:
                                                                                      (|List.length_941/server|
                                                                                        (odds_946/server (|get.::.1_934/server| l_952/server)))
                                                                                      expansions:
                                                                                      • unroll
                                                                                        expr:
                                                                                        (|List.length_941/server| l_952/server)
                                                                                        expansions:
                                                                                        • unroll
                                                                                          expr:
                                                                                          (odds_946/server (|get.::.1_934/server| l_952/server))
                                                                                          expansions:
                                                                                          • Unsat

                                                                                          Let's experiment a bit with merge_sort to gain confidence we've defined it correctly.

                                                                                          In [12]:
                                                                                          merge_sort [1;2;3]
                                                                                          
                                                                                          Out[12]:
                                                                                          - : Z.t list = [1;2;3]
                                                                                          
                                                                                          In [13]:
                                                                                          merge_sort [9;100;6;2;34;19;3;4;7;6]
                                                                                          
                                                                                          Out[13]:
                                                                                          - : Z.t list = [2;3;4;6;6;7;9;19;34;100]
                                                                                          

                                                                                          This looks pretty good! Let's now use Imandra to prove it correct.

                                                                                          Proving Merge Sort correct

                                                                                          What does it mean for a sorting algorithm to be correct? There are two main components:

                                                                                          • The result is sorted
                                                                                          • The result has the same elements as the original

                                                                                          Let's start by defining what it means for a list to be sorted.

                                                                                          In [14]:
                                                                                          let rec is_sorted (x : int list) =
                                                                                           match x with
                                                                                            | []  -> true
                                                                                            | [_] -> true
                                                                                            | x :: x' :: xs -> x <= x' && is_sorted (x' :: xs)
                                                                                          
                                                                                          Out[14]:
                                                                                          val is_sorted : Z.t list -> bool = <fun>
                                                                                          
                                                                                          termination proof

                                                                                          Termination proof

                                                                                          call `let (_x_0 : int list) = List.tl x in is_sorted ((List.hd _x_0) :: (List.tl _x_0))` from `is_sorted x`
                                                                                          original:is_sorted x
                                                                                          sub:let (_x_0 : int list) = List.tl x in is_sorted ((List.hd _x_0) :: (List.tl _x_0))
                                                                                          original ordinal:Ordinal.Int (_cnt x)
                                                                                          sub ordinal:let (_x_0 : int list) = List.tl x in Ordinal.Int (_cnt ((List.hd _x_0) :: (List.tl _x_0)))
                                                                                          path:[List.hd x <= List.hd (List.tl x) && (List.tl x) <> [] && x <> []]
                                                                                          proof:
                                                                                          detailed proof
                                                                                          ground_instances:3
                                                                                          definitions:0
                                                                                          inductions:0
                                                                                          search_time:
                                                                                          0.008s
                                                                                          details:
                                                                                          Expand
                                                                                          smt_stats:
                                                                                          num checks:8
                                                                                          arith assert lower:19
                                                                                          arith tableau max rows:10
                                                                                          arith tableau max columns:24
                                                                                          arith pivots:11
                                                                                          rlimit count:2817
                                                                                          mk clause:31
                                                                                          datatype occurs check:12
                                                                                          mk bool var:87
                                                                                          arith assert upper:23
                                                                                          datatype splits:1
                                                                                          decisions:17
                                                                                          arith row summations:18
                                                                                          arith bound prop:1
                                                                                          propagations:26
                                                                                          interface eqs:1
                                                                                          conflicts:10
                                                                                          arith fixed eqs:6
                                                                                          arith assume eqs:1
                                                                                          datatype accessor ax:10
                                                                                          arith conflicts:3
                                                                                          arith num rows:10
                                                                                          datatype constructor ax:8
                                                                                          num allocs:3080340687
                                                                                          final checks:5
                                                                                          added eqs:52
                                                                                          del clause:22
                                                                                          arith eq adapter:17
                                                                                          memory:20.320000
                                                                                          max memory:27.090000
                                                                                          Expand
                                                                                          • start[0.008s]
                                                                                              let (_x_0 : int list) = List.tl x in
                                                                                              let (_x_1 : int) = List.hd _x_0 in
                                                                                              let (_x_2 : int) = count.list mk_nat x in
                                                                                              let (_x_3 : int) = count.list mk_nat (_x_1 :: …) in
                                                                                              (List.hd x <= _x_1)
                                                                                              && (_x_0 <> [] && (x <> [] && ((_x_2 >= 0) && (_x_3 >= 0))))
                                                                                              ==> not ((_x_1 <= List.hd …) && … <> [])
                                                                                                  || Ordinal.( << ) (Ordinal.Int _x_3) (Ordinal.Int _x_2)
                                                                                          • simplify
                                                                                            into:
                                                                                            let (_x_0 : int list) = List.tl x in
                                                                                            let (_x_1 : int) = List.hd _x_0 in
                                                                                            let (_x_2 : int) = count.list mk_nat (_x_1 :: …) in
                                                                                            let (_x_3 : int) = count.list mk_nat x in
                                                                                            not ((_x_1 <= List.hd …) && … <> [])
                                                                                            || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_3)
                                                                                            || not
                                                                                               ((List.hd x <= _x_1) && _x_0 <> [] && x <> [] && (_x_3 >= 0)
                                                                                                && (_x_2 >= 0))
                                                                                            expansions:
                                                                                            []
                                                                                            rewrite_steps:
                                                                                              forward_chaining:
                                                                                              • unroll
                                                                                                expr:
                                                                                                (let ((a!1 (|count.list_1044/server|
                                                                                                             (|::| (|get.::.0_1031/server|
                                                                                                                    …
                                                                                                expansions:
                                                                                                • unroll
                                                                                                  expr:
                                                                                                  (|count.list_1044/server|
                                                                                                    (|::| (|get.::.0_1031/server| (|get.::.1_1032/server| x_1033/server))
                                                                                                    …
                                                                                                  expansions:
                                                                                                  • unroll
                                                                                                    expr:
                                                                                                    (|count.list_1044/server| x_1033/server)
                                                                                                    expansions:
                                                                                                    • Unsat

                                                                                                    As usual, let's experiment a bit with this definition.

                                                                                                    In [15]:
                                                                                                    is_sorted [1;2;3;4;5]
                                                                                                    
                                                                                                    Out[15]:
                                                                                                    - : bool = true
                                                                                                    
                                                                                                    In [16]:
                                                                                                    is_sorted [1;4;2]
                                                                                                    
                                                                                                    Out[16]:
                                                                                                    - : bool = false
                                                                                                    
                                                                                                    In [17]:
                                                                                                    is_sorted [1]
                                                                                                    
                                                                                                    Out[17]:
                                                                                                    - : bool = true
                                                                                                    
                                                                                                    In [18]:
                                                                                                    instance (fun x -> List.length x >= 5 && is_sorted x)
                                                                                                    
                                                                                                    Out[18]:
                                                                                                    - : Z.t list -> bool = <fun>
                                                                                                    module CX : sig val x : Z.t list end
                                                                                                    
                                                                                                    Instance (after 11 steps, 0.012s):
                                                                                                    let x : int list = [0; 1; 1237; 3033; 3482]
                                                                                                    
                                                                                                    Instance
                                                                                                    proof attempt
                                                                                                    ground_instances:11
                                                                                                    definitions:0
                                                                                                    inductions:0
                                                                                                    search_time:
                                                                                                    0.012s
                                                                                                    details:
                                                                                                    Expand
                                                                                                    smt_stats:
                                                                                                    arith offset eqs:2
                                                                                                    num checks:23
                                                                                                    arith assert lower:57
                                                                                                    arith tableau max rows:18
                                                                                                    arith tableau max columns:34
                                                                                                    arith pivots:26
                                                                                                    rlimit count:7292
                                                                                                    mk clause:103
                                                                                                    datatype occurs check:71
                                                                                                    mk bool var:324
                                                                                                    arith assert upper:43
                                                                                                    datatype splits:21
                                                                                                    decisions:94
                                                                                                    arith row summations:66
                                                                                                    arith bound prop:2
                                                                                                    propagations:173
                                                                                                    conflicts:29
                                                                                                    arith fixed eqs:24
                                                                                                    datatype accessor ax:16
                                                                                                    minimized lits:4
                                                                                                    arith conflicts:8
                                                                                                    arith num rows:18
                                                                                                    arith assert diseq:27
                                                                                                    datatype constructor ax:64
                                                                                                    num allocs:3188283102
                                                                                                    final checks:29
                                                                                                    added eqs:324
                                                                                                    del clause:78
                                                                                                    arith eq adapter:51
                                                                                                    memory:20.550000
                                                                                                    max memory:27.090000
                                                                                                    Expand
                                                                                                    • start[0.012s] (List.length ( :var_0: ) >= 5) && is_sorted ( :var_0: )
                                                                                                    • unroll
                                                                                                      expr:
                                                                                                      (is_sorted_1282/client x_1289/client)
                                                                                                      expansions:
                                                                                                      • unroll
                                                                                                        expr:
                                                                                                        (|List.length_1085/server| x_1289/client)
                                                                                                        expansions:
                                                                                                        • unroll
                                                                                                          expr:
                                                                                                          (|List.length_1085/server| (|get.::.1_1084/server| x_1289/client))
                                                                                                          expansions:
                                                                                                          • unroll
                                                                                                            expr:
                                                                                                            (is_sorted_1282/client
                                                                                                              (|::| (|get.::.0_1083/server| (|get.::.1_1084/server| x_1289/client))
                                                                                                                 …
                                                                                                            expansions:
                                                                                                            • unroll
                                                                                                              expr:
                                                                                                              (|List.length_1085/server|
                                                                                                                (|get.::.1_1084/server| (|get.::.1_1084/server| x_1289/client)))
                                                                                                              expansions:
                                                                                                              • unroll
                                                                                                                expr:
                                                                                                                (let ((a!1 (|::| (|get.::.0_1083/server|
                                                                                                                                   (|get.::.1_1084/server|
                                                                                                                                …
                                                                                                                expansions:
                                                                                                                • unroll
                                                                                                                  expr:
                                                                                                                  (|List.length_1085/server|
                                                                                                                    (|get.::.1_1084/server|
                                                                                                                      (|get.::.1_1084/server| (|get.::.1_1084/ser…
                                                                                                                  expansions:
                                                                                                                  • unroll
                                                                                                                    expr:
                                                                                                                    (let ((a!1 (|get.::.1_1084/server|
                                                                                                                                 (|get.::.1_1084/server|
                                                                                                                                   (|get.::.1_10…
                                                                                                                    expansions:
                                                                                                                    • unroll
                                                                                                                      expr:
                                                                                                                      (let ((a!1 (|get.::.0_1083/server|
                                                                                                                                   (|get.::.1_1084/server|
                                                                                                                                     (|get.::.1_10…
                                                                                                                      expansions:
                                                                                                                      • unroll
                                                                                                                        expr:
                                                                                                                        (let ((a!1 (|get.::.1_1084/server|
                                                                                                                                     (|get.::.1_1084/server|
                                                                                                                                       (|get.::.1_10…
                                                                                                                        expansions:
                                                                                                                        • unroll
                                                                                                                          expr:
                                                                                                                          (let ((a!1 (|get.::.1_1084/server|
                                                                                                                                       (|get.::.1_1084/server|
                                                                                                                                         (|get.::.1_10…
                                                                                                                          expansions:
                                                                                                                          • Sat (Some let x : int list = [(Z.of_nativeint (0n)); (Z.of_nativeint (1n)); (Z.of_nativeint (1237n)); (Z.of_nativeint (3033n)); (Z.of_nativeint (3482n))] )
                                                                                                                          In [19]:
                                                                                                                          instance (fun x -> List.length x >= 5 && is_sorted x && is_sorted (List.rev x))
                                                                                                                          
                                                                                                                          Out[19]:
                                                                                                                          - : Z.t list -> bool = <fun>
                                                                                                                          module CX : sig val x : Z.t list end
                                                                                                                          
                                                                                                                          Instance (after 44 steps, 0.047s):
                                                                                                                          let x : int list = [3634; 3634; 3634; 3634; 3634]
                                                                                                                          
                                                                                                                          Instance
                                                                                                                          proof attempt
                                                                                                                          ground_instances:44
                                                                                                                          definitions:0
                                                                                                                          inductions:0
                                                                                                                          search_time:
                                                                                                                          0.047s
                                                                                                                          details:
                                                                                                                          Expand
                                                                                                                          smt_stats:
                                                                                                                          arith offset eqs:441
                                                                                                                          num checks:89
                                                                                                                          arith assert lower:510
                                                                                                                          arith tableau max rows:45
                                                                                                                          arith tableau max columns:69
                                                                                                                          arith pivots:230
                                                                                                                          rlimit count:95999
                                                                                                                          mk clause:728
                                                                                                                          datatype occurs check:1982
                                                                                                                          mk bool var:3175
                                                                                                                          arith assert upper:700
                                                                                                                          datatype splits:425
                                                                                                                          decisions:2894
                                                                                                                          arith row summations:1420
                                                                                                                          arith bound prop:54
                                                                                                                          propagations:4833
                                                                                                                          interface eqs:56
                                                                                                                          conflicts:186
                                                                                                                          arith fixed eqs:447
                                                                                                                          arith assume eqs:56
                                                                                                                          datatype accessor ax:364
                                                                                                                          minimized lits:25
                                                                                                                          arith conflicts:10
                                                                                                                          arith num rows:45
                                                                                                                          arith assert diseq:202
                                                                                                                          datatype constructor ax:962
                                                                                                                          num allocs:3322874964
                                                                                                                          final checks:192
                                                                                                                          added eqs:8843
                                                                                                                          del clause:463
                                                                                                                          arith eq adapter:266
                                                                                                                          time:0.001000
                                                                                                                          memory:21.940000
                                                                                                                          max memory:27.090000
                                                                                                                          Expand
                                                                                                                          • start[0.047s]
                                                                                                                              (List.length ( :var_0: ) >= 5)
                                                                                                                              && (is_sorted ( :var_0: ) && is_sorted (List.rev ( :var_0: )))
                                                                                                                          • simplify

                                                                                                                            into:
                                                                                                                            (List.length ( :var_0: ) >= 5) && is_sorted ( :var_0: )
                                                                                                                            && is_sorted (List.rev ( :var_0: ))
                                                                                                                            expansions:
                                                                                                                            []
                                                                                                                            rewrite_steps:
                                                                                                                              forward_chaining:
                                                                                                                              • unroll
                                                                                                                                expr:
                                                                                                                                (is_sorted_1282/client (|List.rev_1128/server| x_1291/client))
                                                                                                                                expansions:
                                                                                                                                • unroll
                                                                                                                                  expr:
                                                                                                                                  (|List.rev_1128/server| x_1291/client)
                                                                                                                                  expansions:
                                                                                                                                  • unroll
                                                                                                                                    expr:
                                                                                                                                    (is_sorted_1282/client x_1291/client)
                                                                                                                                    expansions:
                                                                                                                                    • unroll
                                                                                                                                      expr:
                                                                                                                                      (|List.length_1124/server| x_1291/client)
                                                                                                                                      expansions:
                                                                                                                                      • unroll
                                                                                                                                        expr:
                                                                                                                                        (let ((a!1 (|::| (|get.::.0_1122/server|
                                                                                                                                                           (|get.::.1_1123/server|
                                                                                                                                                        …
                                                                                                                                        expansions:
                                                                                                                                        • unroll
                                                                                                                                          expr:
                                                                                                                                          (|List.append_1132/server|
                                                                                                                                            (|List.rev_1128/server| (|get.::.1_1123/server| x_1291/client))
                                                                                                                                            (|::|…
                                                                                                                                          expansions:
                                                                                                                                          • unroll
                                                                                                                                            expr:
                                                                                                                                            (|List.rev_1128/server| (|get.::.1_1123/server| x_1291/client))
                                                                                                                                            expansions:
                                                                                                                                            • unroll
                                                                                                                                              expr:
                                                                                                                                              (|List.length_1124/server| (|get.::.1_1123/server| x_1291/client))
                                                                                                                                              expansions:
                                                                                                                                              • unroll
                                                                                                                                                expr:
                                                                                                                                                (is_sorted_1282/client
                                                                                                                                                  (|::| (|get.::.0_1122/server| (|get.::.1_1123/server| x_1291/client))
                                                                                                                                                     …
                                                                                                                                                expansions:
                                                                                                                                                • unroll
                                                                                                                                                  expr:
                                                                                                                                                  (let ((a!1 (|get.::.0_1122/server|
                                                                                                                                                               (|get.::.1_1123/server|
                                                                                                                                                                 (|get.::.1_11…
                                                                                                                                                  expansions:
                                                                                                                                                  • unroll
                                                                                                                                                    expr:
                                                                                                                                                    (|List.append_1132/server|
                                                                                                                                                      (|List.rev_1128/server|
                                                                                                                                                        (|get.::.1_1123/server| (|get.::.1_1123/ser…
                                                                                                                                                    expansions:
                                                                                                                                                    • unroll
                                                                                                                                                      expr:
                                                                                                                                                      (|List.rev_1128/server|
                                                                                                                                                        (|get.::.1_1123/server| (|get.::.1_1123/server| x_1291/client)))
                                                                                                                                                      expansions:
                                                                                                                                                      • unroll
                                                                                                                                                        expr:
                                                                                                                                                        (|List.length_1124/server|
                                                                                                                                                          (|get.::.1_1123/server| (|get.::.1_1123/server| x_1291/client)))
                                                                                                                                                        expansions:
                                                                                                                                                        • unroll
                                                                                                                                                          expr:
                                                                                                                                                          (|List.append_1132/server|
                                                                                                                                                            (|get.::.1_1123/server|
                                                                                                                                                              (|List.rev_1128/server| (|get.::.1_1123/ser…
                                                                                                                                                          expansions:
                                                                                                                                                          • unroll
                                                                                                                                                            expr:
                                                                                                                                                            (let ((a!1 (|::| (|get.::.0_1122/server|
                                                                                                                                                                               (|get.::.1_1123/server|
                                                                                                                                                                            …
                                                                                                                                                            expansions:
                                                                                                                                                            • unroll
                                                                                                                                                              expr:
                                                                                                                                                              (let ((a!1 (|List.rev_1128/server|
                                                                                                                                                                           (|get.::.1_1123/server|
                                                                                                                                                                             (|get.::.1_11…
                                                                                                                                                              expansions:
                                                                                                                                                              • unroll
                                                                                                                                                                expr:
                                                                                                                                                                (|List.rev_1128/server|
                                                                                                                                                                  (|get.::.1_1123/server|
                                                                                                                                                                    (|get.::.1_1123/server| (|get.::.1_1123/server…
                                                                                                                                                                expansions:
                                                                                                                                                                • unroll
                                                                                                                                                                  expr:
                                                                                                                                                                  (|List.length_1124/server|
                                                                                                                                                                    (|get.::.1_1123/server|
                                                                                                                                                                      (|get.::.1_1123/server| (|get.::.1_1123/ser…
                                                                                                                                                                  expansions:
                                                                                                                                                                  • unroll
                                                                                                                                                                    expr:
                                                                                                                                                                    (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                 (|List.rev_1128/server|
                                                                                                                                                                                   (|get.::.1_11…
                                                                                                                                                                    expansions:
                                                                                                                                                                    • unroll
                                                                                                                                                                      expr:
                                                                                                                                                                      (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                   (|get.::.1_1123/server|
                                                                                                                                                                                     (|get.::.1_11…
                                                                                                                                                                      expansions:
                                                                                                                                                                      • unroll
                                                                                                                                                                        expr:
                                                                                                                                                                        (let ((a!1 (|get.::.0_1122/server|
                                                                                                                                                                                     (|get.::.1_1123/server|
                                                                                                                                                                                       (|get.::.1_11…
                                                                                                                                                                        expansions:
                                                                                                                                                                        • unroll
                                                                                                                                                                          expr:
                                                                                                                                                                          (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                       (|get.::.1_1123/server|
                                                                                                                                                                                         (|get.::.1_11…
                                                                                                                                                                          expansions:
                                                                                                                                                                          • unroll
                                                                                                                                                                            expr:
                                                                                                                                                                            (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                         (|get.::.1_1123/server|
                                                                                                                                                                                           (|get.::.1_11…
                                                                                                                                                                            expansions:
                                                                                                                                                                            • unroll
                                                                                                                                                                              expr:
                                                                                                                                                                              (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                           (|get.::.1_1123/server|
                                                                                                                                                                                             (|get.::.1_11…
                                                                                                                                                                              expansions:
                                                                                                                                                                              • unroll
                                                                                                                                                                                expr:
                                                                                                                                                                                (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                             (|get.::.1_1123/server|
                                                                                                                                                                                               (|List.rev_11…
                                                                                                                                                                                expansions:
                                                                                                                                                                                • unroll
                                                                                                                                                                                  expr:
                                                                                                                                                                                  (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                               (|get.::.1_1123/server|
                                                                                                                                                                                                 (|get.::.1_11…
                                                                                                                                                                                  expansions:
                                                                                                                                                                                  • unroll
                                                                                                                                                                                    expr:
                                                                                                                                                                                    (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                 (|get.::.1_1123/server|
                                                                                                                                                                                                   (|get.::.1_11…
                                                                                                                                                                                    expansions:
                                                                                                                                                                                    • unroll
                                                                                                                                                                                      expr:
                                                                                                                                                                                      (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                   (|get.::.1_1123/server|
                                                                                                                                                                                                     (|get.::.1_11…
                                                                                                                                                                                      expansions:
                                                                                                                                                                                      • unroll
                                                                                                                                                                                        expr:
                                                                                                                                                                                        (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                     (|get.::.1_1123/server|
                                                                                                                                                                                                       (|get.::.1_11…
                                                                                                                                                                                        expansions:
                                                                                                                                                                                        • unroll
                                                                                                                                                                                          expr:
                                                                                                                                                                                          (let ((a!1 (|List.rev_1128/server|
                                                                                                                                                                                                       (|get.::.1_1123/server|
                                                                                                                                                                                                         (|get.::.1_11…
                                                                                                                                                                                          expansions:
                                                                                                                                                                                          • unroll
                                                                                                                                                                                            expr:
                                                                                                                                                                                            (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                         (|get.::.1_1123/server|
                                                                                                                                                                                                           (|get.::.1_11…
                                                                                                                                                                                            expansions:
                                                                                                                                                                                            • unroll
                                                                                                                                                                                              expr:
                                                                                                                                                                                              (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                           (|List.rev_1128/server|
                                                                                                                                                                                                             (|get.::.1_11…
                                                                                                                                                                                              expansions:
                                                                                                                                                                                              • unroll
                                                                                                                                                                                                expr:
                                                                                                                                                                                                (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                             (|get.::.1_1123/server|
                                                                                                                                                                                                               (|get.::.1_11…
                                                                                                                                                                                                expansions:
                                                                                                                                                                                                • unroll
                                                                                                                                                                                                  expr:
                                                                                                                                                                                                  (let ((a!1 (|List.rev_1128/server|
                                                                                                                                                                                                               (|get.::.1_1123/server|
                                                                                                                                                                                                                 (|get.::.1_11…
                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                    expr:
                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                 (|get.::.1_1123/server|
                                                                                                                                                                                                                   (|get.::.1_11…
                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                      expr:
                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                   (|List.rev_1128/server|
                                                                                                                                                                                                                     (|get.::.1_11…
                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                        expr:
                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                     (|get.::.1_1123/server|
                                                                                                                                                                                                                       (|List.rev_11…
                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                          expr:
                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                       (|List.rev_1128/server|
                                                                                                                                                                                                                         (|get.::.1_11…
                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                            expr:
                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                         (|get.::.1_1123/server|
                                                                                                                                                                                                                           (|get.::.1_11…
                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                              expr:
                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                           (|get.::.1_1123/server|
                                                                                                                                                                                                                             (|get.::.1_11…
                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                             (|get.::.1_1123/server|
                                                                                                                                                                                                                               (|get.::.1_11…
                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                               (|get.::.1_1123/server|
                                                                                                                                                                                                                                 (|get.::.1_11…
                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                                 (|get.::.1_1123/server|
                                                                                                                                                                                                                                   (|get.::.1_11…
                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1123/server|
                                                                                                                                                                                                                                   (|get.::.1_1123/server|
                                                                                                                                                                                                                                     (|List.rev_11…
                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                      • Sat (Some let x : int list = [(Z.of_nativeint (3634n)); (Z.of_nativeint (3634n)); (Z.of_nativeint (3634n)); (Z.of_nativeint (3634n)); (Z.of_nativeint (3634n))] )

                                                                                                                                                                                                                      Proving Merge Sort sorts

                                                                                                                                                                                                                      Now that we've defined our is_sorted predicate, we're ready to state one of our main verification goals: that merge_sort sorts!

                                                                                                                                                                                                                      We can write this this way:

                                                                                                                                                                                                                      theorem merge_sort_sorts x =
                                                                                                                                                                                                                       is_sorted (merge_sort x)
                                                                                                                                                                                                                      

                                                                                                                                                                                                                      Before we try to prove this for all possible cases by induction, let's ask Imandra to verify that this property holds up to our current recursion unrolling bound (100).

                                                                                                                                                                                                                      In [20]:
                                                                                                                                                                                                                      verify (fun x -> is_sorted (merge_sort x))
                                                                                                                                                                                                                      
                                                                                                                                                                                                                      Out[20]:
                                                                                                                                                                                                                      - : Z.t list -> bool = <fun>
                                                                                                                                                                                                                      
                                                                                                                                                                                                                      Unknown (Verified up to bound 100)
                                                                                                                                                                                                                      Expand
                                                                                                                                                                                                                      expanded:
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds …) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds …)) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl x))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = List.tl (List.tl (merge_sort x)) in is_sorted ((List.hd _x_0) :: (List.tl _x_0))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds x) in let (_x_1 : int list) = merge_sort (odds (List.tl x)) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in merge (if _x_2 then List.tl _x_0 else _x_0) (if _x_2 then _x_1 else (List.tl _x_1))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds x)))
                                                                                                                                                                                                                      • odds (odds (List.tl x))
                                                                                                                                                                                                                      • odds (odds (odds …))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • odds (List.tl (odds …))
                                                                                                                                                                                                                      • merge_sort (odds x)
                                                                                                                                                                                                                      • odds (odds …)
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds x) in let (_x_1 : int list) = merge_sort (odds (List.tl x)) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in let (_x_3 : int list) = if _x_2 then List.tl _x_0 else _x_0 in let (_x_4 : int list) = if _x_2 then _x_1 else List.tl _x_1 in let (_x_5 : bool) = not (List.hd _x_4 <= List.hd _x_3) && _x_4 <> [] && _x_3 <> [] in merge (if _x_5 then List.tl _x_3 else _x_3) (if _x_5 then _x_4 else (List.tl _x_4))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds …)))
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl (odds (List.tl (odds x))))))
                                                                                                                                                                                                                      • odds x
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds …))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds (List.tl (odds x)))))))
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl x)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds x in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in merge (if _x_6 then List.tl _x_4 else _x_4) (if _x_6 then _x_5 else (List.tl _x_5))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (odds (List.tl …)))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds …))))
                                                                                                                                                                                                                      • odds (odds …)
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl (odds x))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = List.tl (List.tl (List.tl (merge_sort x))) in is_sorted ((List.hd _x_0) :: (List.tl _x_0))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (List.tl (odds (List.tl …))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds (List.tl (odds …))) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (List.tl (odds …))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • odds (odds x)
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl x) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl x)))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds …)))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds (List.tl (odds …))))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl …) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in merge (if _x_6 then List.tl _x_4 else _x_4) (if _x_6 then _x_5 else (List.tl _x_5))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (odds x)))))
                                                                                                                                                                                                                      • is_sorted (merge_sort x)
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds x))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl x)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (odds x))))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (List.tl x)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds x) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds x))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl x))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds (odds (List.tl …)))) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (odds (odds (List.tl x)))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds x)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds x)) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds x) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • odds (List.tl (odds x))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (List.tl (odds x)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds (odds (List.tl (odds (List.tl x))))) in let (_x_1 : int list) = merge_sort (odds (List.tl (odds (List.tl (odds (List.tl …)))))) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in merge (if _x_2 then List.tl _x_0 else _x_0) (if _x_2 then _x_1 else (List.tl _x_1))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (odds x))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (List.tl (odds (List.tl …)))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (merge_sort …)))) in is_sorted ((List.hd _x_0) :: (List.tl _x_0))
                                                                                                                                                                                                                      • merge (merge_sort (odds (odds (List.tl (odds (List.tl x)))))) (merge_sort (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds …) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds x)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl x) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • odds (List.tl (List.tl x))
                                                                                                                                                                                                                      • merge (merge_sort (odds x)) (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds …)))
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (odds x))))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (List.tl x))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (odds (List.tl …)))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds x)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (odds (List.tl x))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds (odds x))) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • merge_sort (odds (odds x))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds …)))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (odds x)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds x in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • let (_x_0 : int list) = List.tl (merge_sort x) in is_sorted ((List.hd _x_0) :: (List.tl _x_0))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (odds x))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds x) in let (_x_1 : int list) = merge_sort … in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in let (_x_3 : int list) = if _x_2 then List.tl _x_0 else _x_0 in let (_x_4 : int list) = if _x_2 then _x_1 else List.tl _x_1 in let (_x_5 : bool) = not (List.hd _x_4 <= List.hd _x_3) && _x_4 <> [] && _x_3 <> [] in let (_x_6 : int list) = if _x_5 then List.tl _x_3 else _x_3 in let (_x_7 : int list) = if _x_5 then _x_4 else List.tl _x_4 in let (_x_8 : bool) = not (List.hd _x_7 <= List.hd _x_6) && _x_7 <> [] && _x_6 <> [] in merge (if _x_8 then List.tl _x_6 else _x_6) (if _x_8 then _x_7 else (List.tl _x_7))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds (List.tl …)) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (List.tl (odds x))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (List.tl x)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds x in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (List.tl x))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl …)))))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds x)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (odds (List.tl …))))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl x))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds (List.tl x)))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds x)))))
                                                                                                                                                                                                                      • odds (List.tl x)
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (List.tl x))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds x))))
                                                                                                                                                                                                                      • odds (odds (odds (odds x)))
                                                                                                                                                                                                                      • odds (odds (odds x))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl x)))))
                                                                                                                                                                                                                      • merge_sort x
                                                                                                                                                                                                                      blocked:
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (List.tl (odds (odds x)))))))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (List.tl (odds (odds x))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x)))))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (odds (odds (List.tl x))))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in merge (if _x_6 then List.tl _x_4 else _x_4) (if _x_6 then _x_5 else (List.tl _x_5))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (odds …)))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (odds …))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (List.tl (odds (List.tl (odds …)))))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds (odds (odds x)))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds (List.tl (odds (odds x)))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds …) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds (List.tl (odds …)))) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort _x_0 in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (odds …))))
                                                                                                                                                                                                                      • odds (odds (odds (odds …)))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (odds (odds …))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (odds (odds x)))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (List.tl (odds (odds x))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds …) in merge (merge_sort _x_0) (merge_sort (odds (List.tl (odds _x_0))))
                                                                                                                                                                                                                      • merge (merge_sort (odds (odds …))) (merge_sort (odds (List.tl (odds (List.tl (odds (odds (odds x))))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds (odds (odds (odds x)))) in let (_x_1 : int list) = merge_sort (odds …) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in merge (if _x_2 then List.tl _x_0 else _x_0) (if _x_2 then _x_1 else (List.tl _x_1))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (odds (odds x))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (List.tl (odds (odds x))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (odds …))
                                                                                                                                                                                                                      • odds (List.tl (odds …))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds (odds (List.tl (odds (List.tl x))))) in let (_x_1 : int list) = merge_sort (odds (List.tl (odds …))) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in let (_x_3 : int list) = if _x_2 then List.tl _x_0 else _x_0 in let (_x_4 : int list) = if _x_2 then _x_1 else List.tl _x_1 in let (_x_5 : bool) = not (List.hd _x_4 <= List.hd _x_3) && _x_4 <> [] && _x_3 <> [] in merge (if _x_5 then List.tl _x_3 else _x_3) (if _x_5 then _x_4 else (List.tl _x_4))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (odds …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in merge (merge_sort (odds (odds (List.tl (odds _x_0))))) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (List.tl (odds …))
                                                                                                                                                                                                                      • odds (odds …)
                                                                                                                                                                                                                      • odds (odds …)
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (odds x)))))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl …))
                                                                                                                                                                                                                      • odds (List.tl (List.tl …))
                                                                                                                                                                                                                      • let (_x_0 : int list) = List.tl (List.tl (List.tl (List.tl (List.tl (merge_sort x))))) in is_sorted ((List.hd _x_0) :: (List.tl _x_0))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (odds (odds x)))))
                                                                                                                                                                                                                      • odds (odds (odds (List.tl (odds (odds x)))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds …)))
                                                                                                                                                                                                                      • merge_sort (odds (odds …))
                                                                                                                                                                                                                      • merge_sort (odds (odds …))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds …)) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (odds (List.tl (odds …)))))))
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl (odds (List.tl (odds …))))))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (odds (List.tl (odds …))))))
                                                                                                                                                                                                                      • odds (odds (odds (odds …)))
                                                                                                                                                                                                                      • odds (odds (odds (List.tl (odds (List.tl (odds …))))))
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl (odds (odds (odds …))))))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds (List.tl (odds (odds …))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds …) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in merge (if _x_6 then List.tl _x_4 else _x_4) (if _x_6 then _x_5 else (List.tl _x_5))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (odds (List.tl (odds …)))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (odds (List.tl (odds …))))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds (odds …)))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (List.tl (odds (odds x)))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds (odds (List.tl (odds (odds …))))) in let (_x_1 : int list) = merge_sort (odds (List.tl (odds (List.tl (odds (odds (List.tl …))))))) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in merge (if _x_2 then List.tl _x_0 else _x_0) (if _x_2 then _x_1 else (List.tl _x_1))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (odds (odds (odds …)))))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds (odds …))))
                                                                                                                                                                                                                      • odds (odds (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (odds (List.tl (odds …)))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (odds (odds (odds …))))))
                                                                                                                                                                                                                      • odds (List.tl (odds (List.tl (odds (odds (List.tl …))))))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds (odds (odds …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds (List.tl (odds (odds x)))) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (odds …))))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = merge_sort (odds x) in let (_x_1 : int list) = merge_sort (odds (List.tl x)) in let (_x_2 : bool) = not (List.hd _x_1 <= List.hd _x_0) && _x_1 <> [] && _x_0 <> [] in let (_x_3 : int list) = if _x_2 then List.tl _x_0 else _x_0 in let (_x_4 : int list) = List.tl _x_3 in let (_x_5 : int list) = if _x_2 then _x_1 else List.tl _x_1 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_3) && _x_5 <> [] && _x_3 <> [] in let (_x_7 : int list) = if _x_6 then _x_4 else _x_3 in let (_x_8 : int list) = List.tl _x_7 in let (_x_9 : int list) = List.tl _x_5 in let (_x_10 : int list) = if _x_6 then _x_5 else _x_9 in let (_x_11 : bool) = not (List.hd _x_10 <= List.hd _x_7) && _x_10 <> [] && _x_7 <> [] in let (_x_12 : int list) = if _x_11 then _x_8 else if _x_6 then _x_4 else … in let (_x_13 : int list) = List.tl _x_10 in let (_x_14 : int list) = if _x_11 then if _x_6 then … else _x_9 else _x_13 in let (_x_15 : bool) = not (List.hd (if _x_11 then _x_10 else _x_13) <= List.hd (if _x_11 then _x_8 else _x_7)) && _x_14 <> [] && _x_12 <> [] in merge (if _x_15 then List.tl _x_12 else _x_12) (if _x_15 then _x_14 else (List.tl _x_14))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds (odds (List.tl …)))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds (List.tl (odds …))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (odds …)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl x) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in let (_x_7 : int list) = if _x_6 then List.tl _x_4 else _x_4 in let (_x_8 : int list) = if _x_6 then _x_5 else List.tl _x_5 in let (_x_9 : bool) = not (List.hd _x_8 <= List.hd _x_7) && _x_8 <> [] && _x_7 <> [] in merge (if _x_9 then List.tl _x_7 else _x_7) (if _x_9 then _x_8 else (List.tl _x_8))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds (odds …))))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (List.tl (odds (List.tl (odds …)))) in merge (merge_sort (odds _x_0)) (merge_sort _x_0)
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl …))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds (odds …)) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (odds (List.tl (odds (List.tl x)))))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl x))))))))
                                                                                                                                                                                                                      • merge_sort (odds (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds (odds …)) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds (List.tl (odds (List.tl (odds …))))) in merge (merge_sort (odds _x_0)) (merge_sort (odds (List.tl _x_0)))
                                                                                                                                                                                                                      • merge (merge_sort (odds (odds (List.tl (odds (List.tl (odds (odds x)))))))) (merge_sort (odds (List.tl (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (odds …))))
                                                                                                                                                                                                                      • odds (List.tl (odds (odds (List.tl (odds (List.tl (odds …)))))))
                                                                                                                                                                                                                      • odds (List.tl (odds …))
                                                                                                                                                                                                                      • merge (merge_sort (odds (odds (odds (odds (odds …)))))) (merge_sort (odds (List.tl (odds (odds (odds (odds x)))))))
                                                                                                                                                                                                                      • odds (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl (List.tl …)))))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in let (_x_7 : int list) = if _x_6 then List.tl _x_4 else _x_4 in let (_x_8 : int list) = if _x_6 then _x_5 else List.tl _x_5 in let (_x_9 : bool) = not (List.hd _x_8 <= List.hd _x_7) && _x_8 <> [] && _x_7 <> [] in merge (if _x_9 then List.tl _x_7 else _x_7) (if _x_9 then _x_8 else (List.tl _x_8))
                                                                                                                                                                                                                      • odds (odds (odds (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      • odds (odds (List.tl (odds (odds (odds …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds (odds …) in let (_x_1 : int list) = merge_sort (odds _x_0) in let (_x_2 : int list) = merge_sort (odds (List.tl _x_0)) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in let (_x_4 : int list) = if _x_3 then List.tl _x_1 else _x_1 in let (_x_5 : int list) = if _x_3 then _x_2 else List.tl _x_2 in let (_x_6 : bool) = not (List.hd _x_5 <= List.hd _x_4) && _x_5 <> [] && _x_4 <> [] in merge (if _x_6 then List.tl _x_4 else _x_4) (if _x_6 then _x_5 else (List.tl _x_5))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (odds …)))))
                                                                                                                                                                                                                      • let (_x_0 : int list) = odds … in let (_x_1 : int list) = merge_sort (odds (odds _x_0)) in let (_x_2 : int list) = merge_sort (odds (List.tl (odds (odds (List.tl _x_0))))) in let (_x_3 : bool) = not (List.hd _x_2 <= List.hd _x_1) && _x_2 <> [] && _x_1 <> [] in merge (if _x_3 then List.tl _x_1 else _x_1) (if _x_3 then _x_2 else (List.tl _x_2))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (odds (List.tl (odds (List.tl (odds …))))))))
                                                                                                                                                                                                                      • odds (odds (odds (odds (odds …))))
                                                                                                                                                                                                                      • merge_sort (odds (List.tl (odds (List.tl (odds …)))))
                                                                                                                                                                                                                      proof attempt
                                                                                                                                                                                                                      ground_instances:100
                                                                                                                                                                                                                      definitions:0
                                                                                                                                                                                                                      inductions:0
                                                                                                                                                                                                                      search_time:
                                                                                                                                                                                                                      1.275s
                                                                                                                                                                                                                      Expand
                                                                                                                                                                                                                      • start[1.275s] is_sorted (merge_sort ( :var_0: ))
                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                        (is_sorted_1282/client (merge_sort_1276/client x_1293/client))
                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                          (merge_sort_1276/client x_1293/client)
                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                            (merge_sort_1276/client (odds_1289/server x_1293/client))
                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                              (merge_1244/client
                                                                                                                                                                                                                                (merge_sort_1276/client (odds_1289/server x_1293/client))
                                                                                                                                                                                                                                (merge_sort_1276/cl…
                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                (let ((a!1 (|::| (|get.::.0_1287/server|
                                                                                                                                                                                                                                                   (|get.::.1_1288/server|
                                                                                                                                                                                                                                                …
                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                  (merge_sort_1276/client
                                                                                                                                                                                                                                    (odds_1289/server (|get.::.1_1288/server| x_1293/client)))
                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                    (odds_1289/server (|get.::.1_1288/server| x_1293/client))
                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                      (odds_1289/server x_1293/client)
                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                        (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                            (|get.::.1_1288/server| x_1293/client)…
                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                          (let ((a!1 (merge_sort_1276/client
                                                                                                                                                                                                                                                       (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                    …
                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                            (let ((a!1 (|get.::.0_1287/server|
                                                                                                                                                                                                                                                         (merge_sort_1276/client
                                                                                                                                                                                                                                                           (odds_1289/se…
                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                              (merge_sort_1276/client
                                                                                                                                                                                                                                                (odds_1289/server (|get.::.1_1288/server| (odds_1289/server x_1293/client)…
                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                (odds_1289/server (|get.::.1_1288/server| (odds_1289/server x_1293/client)))
                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                  (merge_sort_1276/client (odds_1289/server (odds_1289/server x_1293/client)))
                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                    (odds_1289/server (odds_1289/server x_1293/client))
                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                      (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                          (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                    …
                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                        (let ((a!1 (merge_sort_1276/client
                                                                                                                                                                                                                                                                     (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                          (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                         (odds_1289/serve…
                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                            (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                (odds_1289/server (|get.::.1_1288/serv…
                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.0_1287/server|
                                                                                                                                                                                                                                                                           (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                             (|get.::.1_12…
                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                (merge_sort_1276/client
                                                                                                                                                                                                                                                                  (odds_1289/server (odds_1289/server (|get.::.1_1288/server| x_1293/client)…
                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                  (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                      (|get.::.1_1288/server| (odds_1289/ser…
                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                    (odds_1289/server (odds_1289/server (|get.::.1_1288/server| x_1293/client)))
                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                      (let ((a!1 (merge_sort_1276/client
                                                                                                                                                                                                                                                                                   (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                …
                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                     (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                       (|get.::.1_12…
                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.0_1287/server|
                                                                                                                                                                                                                                                                                       (merge_sort_1276/client
                                                                                                                                                                                                                                                                                         (odds_1289/se…
                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                         (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                           (|get.::.1_12…
                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                              (let ((a!1 (merge_sort_1276/client
                                                                                                                                                                                                                                                                                           (odds_1289/server (odds_1289/server (odds_1289/serve…
                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                               (odds_1289/serve…
                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                  (let ((a!1 (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                               …
                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                    (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                        (odds_1289/server (odds_1289/server x_…
                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                      (merge_sort_1276/client
                                                                                                                                                                                                                                                                                        (odds_1289/server (odds_1289/server (odds_1289/server x_1293/client))))
                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                        (odds_1289/server (odds_1289/server (odds_1289/server x_1293/client)))
                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                       (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                         (|get.::.1_12…
                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                         (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                      …
                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                           (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                        …
                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                (let ((a!1 (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                             …
                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                  (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                        (o…
                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                   (|get.::.1_12…
                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                      (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                     (odds_1289/serve…
                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                     (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                       (odds_1289/se…
                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                       (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                         (odds_1289/se…
                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                            (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                           (odds_1289/serve…
                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                              (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                             (odds_1289/serve…
                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                               (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                  (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                 (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                    (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                   (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                   (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                     (|get.::.1_12…
                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                        (let ((a!1 (odds_1289/server (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                   …
                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                       (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                         (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                              (let ((a!1 (odds_1289/server (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                         …
                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                (odds_1289/server (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                      …
                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (merge_sort_1276/client
                                                                                                                                                                                                                                                                                                                                               (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                            …
                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_12…
                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.0_1287/server|
                                                                                                                                                                                                                                                                                                                                                   (merge_sort_1276/client
                                                                                                                                                                                                                                                                                                                                                     (odds_1289/se…
                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                       (odds_1289/se…
                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                         (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                           (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_12…
                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                               (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                 (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                   (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                     (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                     (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                  …
                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                         (odds_1289/se…
                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (odds_1289/server (odds_1289/server (odds_1289/server (odds_1289/server x_1293/client))))…
                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                           (odds_1289/server (odds_1289/server (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                             (odds_1289/server (odds_1289/server (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (odds_1289/server (odds_1289/server (odds_1289/server (odds_1289/server x_1293/client))))…
                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                    (odds_1289/server (odds_1289/server (odds_1289/server (odds_1289/server x_1293/client))))
                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                   (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                …
                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                     (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                  …
                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                       (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                    …
                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                         (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                           (|get.::.1_12…
                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                           (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                        …
                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                             (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                          …
                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                               (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                            …
                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                   (|get.::.1_12…
                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                   (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                …
                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                       (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                       (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                         (odds_1289/se…
                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                           (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                             (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                             …
                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                 (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                   (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                     (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                       (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                       …
                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                           (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                             (odds_1289/serve…
                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                             (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                               (|get.::.1_12…
                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                  (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                               (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                    (let ((a!1 (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                 …
                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                      (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                   (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                        (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                     (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                       (odds_1289/se…
                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                          (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                       (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                            (let ((a!1 (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                         (odds_1289/server (odds_1289/server (|get.::.1_1288/…
                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                              (let ((a!1 (odds_1289/server (odds_1289/server (|get.::.1_1288/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                           …
                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:

                                                                                                                                                                                                                                                                                                                                                                                                                              Excellent. This gives us quite some confidence that this result is true. Let's now try to prove it by induction. We'll want to use an induction principle suggested by the definition of merge_sort. Let's use our usual strategy of #max_induct 1, so we can analyse Imandra's output to help us find needed lemmata.

                                                                                                                                                                                                                                                                                                                                                                                                                              Note, we could prove this theorem directly without any lemmas by using #max_induct 3, but it's good practice to use #max_induct 1 and have Imandra help us develop useful collections of lemmas.

                                                                                                                                                                                                                                                                                                                                                                                                                              In [21]:
                                                                                                                                                                                                                                                                                                                                                                                                                              #max_induct 1
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Out[21]:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              In [22]:
                                                                                                                                                                                                                                                                                                                                                                                                                              theorem merge_sort_sorts x =
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                              [@@induct functional merge_sort]
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Out[22]:
                                                                                                                                                                                                                                                                                                                                                                                                                              val merge_sort_sorts : Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                              Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              is_sorted (merge_sort x).
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              We shall induct according to a scheme derived from merge_sort.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               (not (not Is_a([], List.tl x) && x <> []) ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                               && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                   && (not Is_a([], List.tl x) && (φ (odds (List.tl x)) && φ (odds x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                   ==> φ x).
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 2:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. not Is_a([], List.tl x) && x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              But simplification reduces this to true, using the definitions of is_sorted
                                                                                                                                                                                                                                                                                                                                                                                                                              and merge_sort.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. not Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. is_sorted (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. is_sorted (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              This simplifies, using the definition of merge_sort to:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1':
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. is_sorted (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. is_sorted (merge (merge_sort (odds x)) (merge_sort (odds (List.tl x))))
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              We can eliminate destructors by the following substitution:
                                                                                                                                                                                                                                                                                                                                                                                                                               x -> x1 :: x2
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1'':
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted (merge_sort (odds x2))
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted (merge_sort (odds (x1 :: x2)))
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. is_sorted (merge (merge_sort (odds (x1 :: x2))) (merge_sort (odds x2)))
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. Is_a([], x2)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Candidates for generalization:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               odds (x1 :: x2)
                                                                                                                                                                                                                                                                                                                                                                                                                               odds x2
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1''':
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted (merge_sort gen_2)
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted (merge_sort gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. is_sorted (merge (merge_sort gen_1) (merge_sort gen_2))
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. Is_a([], x2)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Candidates for generalization:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               merge_sort gen_1
                                                                                                                                                                                                                                                                                                                                                                                                                               merge_sort gen_2
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              This produces the modified subgoal:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1'''':
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted gen_4
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted gen_3
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. is_sorted (merge gen_3 gen_4)
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. Is_a([], x2)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Must try induction.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               Aborting proof attempt for merge_sort_sorts.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                  (:def is_sorted)
                                                                                                                                                                                                                                                                                                                                                                                                                                  (:def merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                  (:induct merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Checkpoints:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted gen_4
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted gen_3
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. is_sorted (merge gen_3 gen_4)
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. Is_a([], x2)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Error:
                                                                                                                                                                                                                                                                                                                                                                                                                                Maximum induction depth reached (1). You can set this with #max_induct.
                                                                                                                                                                                                                                                                                                                                                                                                                                At <none>:1
                                                                                                                                                                                                                                                                                                                                                                                                                                

                                                                                                                                                                                                                                                                                                                                                                                                                              Analysing the output of this proof attempt, we notice the following components of our Checkpoint:

                                                                                                                                                                                                                                                                                                                                                                                                                              H1. is_sorted gen_1
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. is_sorted gen_2
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge gen_2 gen_1)
                                                                                                                                                                                                                                                                                                                                                                                                                              

                                                                                                                                                                                                                                                                                                                                                                                                                              Ah, of course! We need to prove a lemma that merge respects the sortedness of its inputs.

                                                                                                                                                                                                                                                                                                                                                                                                                              Let's do this by functional induction following the definition of merge and install it as a rewrite rule. We'll allow nested induction (#max_induct 2), but it's a good exercise to do this proof without it (deriving an additional lemma!).

                                                                                                                                                                                                                                                                                                                                                                                                                              In [23]:
                                                                                                                                                                                                                                                                                                                                                                                                                              #max_induct 2;;
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              theorem merge_sorts x y =
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted x && is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               ==>
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                              [@@induct functional merge]
                                                                                                                                                                                                                                                                                                                                                                                                                              [@@rewrite]
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Out[23]:
                                                                                                                                                                                                                                                                                                                                                                                                                              val merge_sorts : Z.t list -> Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                              Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              is_sorted x && is_sorted y ==> is_sorted (merge x y).
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              We shall induct according to a scheme derived from merge.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               (not ((List.hd x < List.hd y) && (y <> [] && x <> []))
                                                                                                                                                                                                                                                                                                                                                                                                                                && not (not (List.hd x < List.hd y) && (y <> [] && x <> [])) ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                               && ((x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                    && (y <> [] && (not (List.hd x < List.hd y) && φ x (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                                    ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                   && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                       && (y <> [] && ((List.hd x < List.hd y) && φ (List.tl x) y))
                                                                                                                                                                                                                                                                                                                                                                                                                                       ==> φ x y)).
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              3 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 3:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. not (not (List.hd y <= List.hd x) && y <> [] && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. not ((List.hd y <= List.hd x) && y <> [] && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              But simplification reduces this to true, using the definitions of is_sorted
                                                                                                                                                                                                                                                                                                                                                                                                                              and merge.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 2:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H4. List.hd y <= List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                               H5. is_sorted x && is_sorted (List.tl y) ==> is_sorted (merge x (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              This simplifies, using the definition of merge to the following 2 subgoals:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 2.2:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. List.hd y <= List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H4. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. is_sorted (merge x (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. is_sorted (List.tl y)
                                                                                                                                                                                                                                                                                                                                                                                                                               C2. is_sorted ((List.hd y) :: (merge x (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              But we verify Subgoal 2.2 by recursive unrolling.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 2.1:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. List.hd y <= List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H4. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H5. is_sorted (merge x (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted ((List.hd y) :: (merge x (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              But we verify Subgoal 2.1 by recursive unrolling.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H4. not (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                               H5. is_sorted (List.tl x) && is_sorted y ==> is_sorted (merge (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              This simplifies, using the definition of merge to the following 2 subgoals:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. is_sorted (merge (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. is_sorted (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                               C2. List.hd y <= List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                               C3. is_sorted ((List.hd x) :: (merge (List.tl x) y))
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              But we verify Subgoal 1.2 by recursive unrolling.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               H0. is_sorted x
                                                                                                                                                                                                                                                                                                                                                                                                                               H1. is_sorted y
                                                                                                                                                                                                                                                                                                                                                                                                                               H2. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H3. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                               H4. is_sorted (merge (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                              |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                               C0. List.hd y <= List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                               C1. is_sorted ((List.hd x) :: (merge (List.tl x) y))
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              But we verify Subgoal 1.1 by recursive unrolling.
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                               Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                  (:def is_sorted)
                                                                                                                                                                                                                                                                                                                                                                                                                                  (:def merge)
                                                                                                                                                                                                                                                                                                                                                                                                                                  (:induct merge)
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                              Proved
                                                                                                                                                                                                                                                                                                                                                                                                                              proof
                                                                                                                                                                                                                                                                                                                                                                                                                              ground_instances:16
                                                                                                                                                                                                                                                                                                                                                                                                                              definitions:12
                                                                                                                                                                                                                                                                                                                                                                                                                              inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                              search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                              0.894s
                                                                                                                                                                                                                                                                                                                                                                                                                              details:
                                                                                                                                                                                                                                                                                                                                                                                                                              Expand
                                                                                                                                                                                                                                                                                                                                                                                                                              smt_stats:
                                                                                                                                                                                                                                                                                                                                                                                                                              num checks:12
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-make-feasible:30
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-max-columns:19
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-conflicts:4
                                                                                                                                                                                                                                                                                                                                                                                                                              rlimit count:281359
                                                                                                                                                                                                                                                                                                                                                                                                                              mk clause:265
                                                                                                                                                                                                                                                                                                                                                                                                                              datatype occurs check:78
                                                                                                                                                                                                                                                                                                                                                                                                                              mk bool var:353
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-lower:22
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-diseq:2
                                                                                                                                                                                                                                                                                                                                                                                                                              datatype splits:21
                                                                                                                                                                                                                                                                                                                                                                                                                              decisions:93
                                                                                                                                                                                                                                                                                                                                                                                                                              propagations:283
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-max-rows:9
                                                                                                                                                                                                                                                                                                                                                                                                                              conflicts:19
                                                                                                                                                                                                                                                                                                                                                                                                                              datatype accessor ax:18
                                                                                                                                                                                                                                                                                                                                                                                                                              minimized lits:5
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-bound-propagations-lp:1
                                                                                                                                                                                                                                                                                                                                                                                                                              datatype constructor ax:48
                                                                                                                                                                                                                                                                                                                                                                                                                              final checks:14
                                                                                                                                                                                                                                                                                                                                                                                                                              added eqs:540
                                                                                                                                                                                                                                                                                                                                                                                                                              del clause:214
                                                                                                                                                                                                                                                                                                                                                                                                                              arith eq adapter:12
                                                                                                                                                                                                                                                                                                                                                                                                                              arith-upper:18
                                                                                                                                                                                                                                                                                                                                                                                                                              memory:34.430000
                                                                                                                                                                                                                                                                                                                                                                                                                              max memory:44.370000
                                                                                                                                                                                                                                                                                                                                                                                                                              num allocs:48748162091.000000
                                                                                                                                                                                                                                                                                                                                                                                                                              Expand
                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.894s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                  is_sorted ( :var_0: ) && is_sorted ( :var_1: )
                                                                                                                                                                                                                                                                                                                                                                                                                                  ==> is_sorted (merge ( :var_0: ) ( :var_1: ))
                                                                                                                                                                                                                                                                                                                                                                                                                              • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                not (is_sorted x) || not (is_sorted y) || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                • start[0.894s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                    not (is_sorted x) || not (is_sorted y) || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                • induction on (functional merge)
                                                                                                                                                                                                                                                                                                                                                                                                                                  :scheme (not ((List.hd x < List.hd y) && (y <> [] && x <> []))
                                                                                                                                                                                                                                                                                                                                                                                                                                           && not (not (List.hd x < List.hd y) && (y <> [] && x <> []))
                                                                                                                                                                                                                                                                                                                                                                                                                                           ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                          && ((x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                               && (y <> [] && (not (List.hd x < List.hd y) && φ x (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                                               ==> φ x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                              && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                  && (y <> [] && ((List.hd x < List.hd y) && φ (List.tl x) y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                  ==> φ x y))
                                                                                                                                                                                                                                                                                                                                                                                                                                • Split (let (_x_0 : bool) = is_sorted x in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_1 : bool) = is_sorted y in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_2 : bool) = is_sorted (merge x y) || not (_x_0 && _x_1) in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_3 : bool) = List.hd y <= List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_4 : bool) = not _x_3 in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_5 : bool) = y <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_6 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_7 : bool) = _x_6 && _x_5 in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_8 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                         let (_x_9 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                         (_x_2
                                                                                                                                                                                                                                                                                                                                                                                                                                          || not (not (_x_4 && _x_5 && _x_6) && not (_x_3 && _x_5 && _x_6)))
                                                                                                                                                                                                                                                                                                                                                                                                                                         && (_x_2
                                                                                                                                                                                                                                                                                                                                                                                                                                             || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                (_x_7 && _x_3
                                                                                                                                                                                                                                                                                                                                                                                                                                                 && (is_sorted (merge x _x_8) || not (_x_0 && is_sorted _x_8))))
                                                                                                                                                                                                                                                                                                                                                                                                                                         && (_x_2
                                                                                                                                                                                                                                                                                                                                                                                                                                             || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                (_x_7 && _x_4
                                                                                                                                                                                                                                                                                                                                                                                                                                                 && (is_sorted (merge _x_9 y) || not (is_sorted _x_9 && _x_1))))
                                                                                                                                                                                                                                                                                                                                                                                                                                         :cases [let (_x_0 : bool) = List.hd y <= List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 let (_x_1 : bool) = y <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 not (is_sorted x) || not (is_sorted y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || (not _x_0 && _x_1 && _x_2) || (_x_0 && _x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || is_sorted (merge x y);
                                                                                                                                                                                                                                                                                                                                                                                                                                                 let (_x_0 : bool) = is_sorted x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 let (_x_1 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 not _x_0 || not (is_sorted y) || not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || not (y <> []) || not (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || not (_x_0 && is_sorted _x_1 ==> is_sorted (merge x _x_1))
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || is_sorted (merge x y);
                                                                                                                                                                                                                                                                                                                                                                                                                                                 let (_x_0 : bool) = is_sorted y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 let (_x_1 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                 not (is_sorted x) || not _x_0 || not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || not (y <> []) || (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || not (is_sorted _x_1 && _x_0 ==> is_sorted (merge _x_1 y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                 || is_sorted (merge x y)])
                                                                                                                                                                                                                                                                                                                                                                                                                                  • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                    let (_x_0 : bool) = is_sorted y in let (_x_1 : int list) = List.tl x in not (is_sorted x) || not _x_0 || not (x <> []) || not (y <> []) || (List.hd y <= List.hd x) || not (is_sorted _x_1 && _x_0 ==> is_sorted (merge _x_1 y)) || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                    • start[0.894s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : bool) = is_sorted y in
                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_1 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                        not (is_sorted x) || not _x_0 || not (x <> []) || not (y <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                        || (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                        || not (is_sorted _x_1 && _x_0 ==> is_sorted (merge _x_1 y))
                                                                                                                                                                                                                                                                                                                                                                                                                                        || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                    • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                      into:
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_1 : int list) = merge _x_0 y in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_2 : bool) = is_sorted _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_3 : int) = List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_4 : bool) = List.hd y <= _x_3 in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_5 : bool) = not (is_sorted x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_6 : bool) = not (is_sorted y) in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_7 : bool) = not (x <> []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_8 : bool) = not (y <> []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_9 : bool) = is_sorted (_x_3 :: _x_1) in
                                                                                                                                                                                                                                                                                                                                                                                                                                      (_x_2 || is_sorted _x_0 || _x_4 || _x_5 || _x_6 || _x_7 || _x_8 || _x_9)
                                                                                                                                                                                                                                                                                                                                                                                                                                      && (_x_4 || _x_5 || _x_6 || _x_7 || _x_8 || _x_9 || not _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                      [merge, merge]
                                                                                                                                                                                                                                                                                                                                                                                                                                      rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                        forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                          • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                          • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                      • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : bool) = is_sorted x in let (_x_1 : int list) = List.tl y in not _x_0 || not (is_sorted y) || not (x <> []) || not (y <> []) || not (List.hd y <= List.hd x) || not (_x_0 && is_sorted _x_1 ==> is_sorted (merge x _x_1)) || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                        • start[0.894s, "2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_0 : bool) = is_sorted x in
                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_1 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                            not _x_0 || not (is_sorted y) || not (x <> []) || not (y <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                            || not (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                            || not (_x_0 && is_sorted _x_1 ==> is_sorted (merge x _x_1))
                                                                                                                                                                                                                                                                                                                                                                                                                                            || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                        • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                          into:
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_1 : int list) = merge x _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_2 : bool) = is_sorted _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_3 : int) = List.hd y in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_4 : bool) = not (_x_3 <= List.hd x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_5 : bool) = not (is_sorted x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_6 : bool) = not (is_sorted y) in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_7 : bool) = not (x <> []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_8 : bool) = not (y <> []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_9 : bool) = is_sorted (_x_3 :: _x_1) in
                                                                                                                                                                                                                                                                                                                                                                                                                                          (_x_2 || is_sorted _x_0 || _x_4 || _x_5 || _x_6 || _x_7 || _x_8 || _x_9)
                                                                                                                                                                                                                                                                                                                                                                                                                                          && (_x_4 || _x_5 || _x_6 || _x_7 || _x_8 || _x_9 || not _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                          [merge, merge]
                                                                                                                                                                                                                                                                                                                                                                                                                                          rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                            forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                              • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                              • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                          • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_0 : bool) = List.hd y <= List.hd x in let (_x_1 : bool) = y <> [] in let (_x_2 : bool) = x <> [] in not (is_sorted x) || not (is_sorted y) || (not _x_0 && _x_1 && _x_2) || (_x_0 && _x_1 && _x_2) || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                            • start[0.894s, "3"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_0 : bool) = List.hd y <= List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_1 : bool) = y <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                not (is_sorted x) || not (is_sorted y) || (not _x_0 && _x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                || (_x_0 && _x_1 && _x_2) || is_sorted (merge x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                            • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                              into:
                                                                                                                                                                                                                                                                                                                                                                                                                                              true
                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                              [merge, is_sorted, merge, is_sorted, merge, is_sorted, merge, is_sorted]
                                                                                                                                                                                                                                                                                                                                                                                                                                              rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                          Excellent! Now that we have that key lemma proved and available as a rewrite rule, let's return to our original goal:

                                                                                                                                                                                                                                                                                                                                                                                                                                          In [24]:
                                                                                                                                                                                                                                                                                                                                                                                                                                          theorem merge_sort_sorts x =
                                                                                                                                                                                                                                                                                                                                                                                                                                           is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                          [@@induct functional merge_sort]
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          Out[24]:
                                                                                                                                                                                                                                                                                                                                                                                                                                          val merge_sort_sorts : Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                          Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          is_sorted (merge_sort x).
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          We shall induct according to a scheme derived from merge_sort.
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                           (not (not Is_a([], List.tl x) && x <> []) ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                                           && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                               && (not Is_a([], List.tl x) && (φ (odds (List.tl x)) && φ (odds x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                               ==> φ x).
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          Subgoal 2:
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                           C0. not Is_a([], List.tl x) && x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                           C1. is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          But simplification reduces this to true, using the definitions of is_sorted
                                                                                                                                                                                                                                                                                                                                                                                                                                          and merge_sort.
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                           H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                           H1. not Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                           H2. is_sorted (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                           H3. is_sorted (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                          |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                           is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          This simplifies, using the definition of merge_sort to:
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          Subgoal 1':
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                           H0. is_sorted (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                           H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                           H2. is_sorted (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                          |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                           C0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                           C1. is_sorted (merge (merge_sort (odds x)) (merge_sort (odds (List.tl x))))
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          But simplification reduces this to true, using the rewrite rule merge_sorts.
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                           Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                              (:def is_sorted)
                                                                                                                                                                                                                                                                                                                                                                                                                                              (:def merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                              (:rw merge_sorts)
                                                                                                                                                                                                                                                                                                                                                                                                                                              (:induct merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                          Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                          proof
                                                                                                                                                                                                                                                                                                                                                                                                                                          ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                          definitions:5
                                                                                                                                                                                                                                                                                                                                                                                                                                          inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                          search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                          0.194s
                                                                                                                                                                                                                                                                                                                                                                                                                                          Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.194s, "Goal"] is_sorted (merge_sort ( :var_0: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                          • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                            is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                            • start[0.194s, "1"] is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                            • induction on (functional merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                              :scheme (not (not Is_a([], List.tl x) && x <> []) ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                      && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                          && (not Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                              && (φ (odds (List.tl x)) && φ (odds x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                          ==> φ x)
                                                                                                                                                                                                                                                                                                                                                                                                                                            • Split (let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                     let (_x_1 : bool) = not Is_a([], _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                     let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                     let (_x_3 : bool) = is_sorted (merge_sort x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                     ((_x_1 && _x_2) || _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                     && (_x_3
                                                                                                                                                                                                                                                                                                                                                                                                                                                         || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                            (_x_2 && _x_1 && is_sorted (merge_sort (odds _x_0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                             && is_sorted (merge_sort (odds x))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                     :cases [(not Is_a([], List.tl x) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                             || is_sorted (merge_sort x);
                                                                                                                                                                                                                                                                                                                                                                                                                                                             let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                             not (x <> []) || Is_a([], _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                             || not (is_sorted (merge_sort (odds _x_0)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                             || not (is_sorted (merge_sort (odds x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                             || is_sorted (merge_sort x)])
                                                                                                                                                                                                                                                                                                                                                                                                                                              • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_0 : int list) = List.tl x in not (x <> []) || Is_a([], _x_0) || not (is_sorted (merge_sort (odds _x_0))) || not (is_sorted (merge_sort (odds x))) || is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                • start[0.193s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                    let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                    not (x <> []) || Is_a([], _x_0) || not (is_sorted (merge_sort (odds _x_0)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                    || not (is_sorted (merge_sort (odds x))) || is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                  into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_1 : int list) = merge_sort (odds x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_2 : int list) = merge_sort (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                  Is_a([], _x_0) || not (is_sorted _x_1) || not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                  || not (is_sorted _x_2) || is_sorted (merge _x_1 _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                  merge_sort
                                                                                                                                                                                                                                                                                                                                                                                                                                                  rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                      into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      true
                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      []
                                                                                                                                                                                                                                                                                                                                                                                                                                                      rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      merge_sorts
                                                                                                                                                                                                                                                                                                                                                                                                                                                      forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                      (not Is_a([], List.tl x) && x <> []) || is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.193s, "2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                          (not Is_a([], List.tl x) && x <> []) || is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                      • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                        into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        true
                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        [is_sorted, merge_sort, is_sorted, merge_sort]
                                                                                                                                                                                                                                                                                                                                                                                                                                                        rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Beautiful! So now we've proved half of our specification: merge_sort sorts!

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Merge Sort contains the right elements

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Let's now turn to the second half of our correctness criterion: That merge_sort x contains "the same" elements as x.

                                                                                                                                                                                                                                                                                                                                                                                                                                                    What does this mean, and why does it matter?

                                                                                                                                                                                                                                                                                                                                                                                                                                                    An incorrect sorting function

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Consider the sorting function bad_sort defined as bad_sort x = [].

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Clearly, we could prove theorem bad_sort_sorts x = is_sorted (bad_sort x), even though we'd never want to use bad_sort as a sorting function in practice.

                                                                                                                                                                                                                                                                                                                                                                                                                                                    That is, computing a sorted list is only one piece of the puzzle. We must also verify that merge_sort x contains exactly the same elements as x, including their multiplicity).

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Multiset equivalence

                                                                                                                                                                                                                                                                                                                                                                                                                                                    How can we express this concept? We'll use the notion of a multiset and count the number of occurrences of each element of x, and make sure these are respected by merge_sort.

                                                                                                                                                                                                                                                                                                                                                                                                                                                    We'll begin by defining a function num_occurs x y which counts the number of times an element x appears in the list y.

                                                                                                                                                                                                                                                                                                                                                                                                                                                    In [25]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    let rec num_occurs x y =
                                                                                                                                                                                                                                                                                                                                                                                                                                                     match y with
                                                                                                                                                                                                                                                                                                                                                                                                                                                      | [] -> 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                      | hd :: tl when hd = x ->
                                                                                                                                                                                                                                                                                                                                                                                                                                                        1 + num_occurs x tl
                                                                                                                                                                                                                                                                                                                                                                                                                                                      | _ :: tl ->
                                                                                                                                                                                                                                                                                                                                                                                                                                                        num_occurs x tl
                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                    Out[25]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    val num_occurs : 'a -> 'a list -> Z.t = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                    termination proof

                                                                                                                                                                                                                                                                                                                                                                                                                                                    Termination proof

                                                                                                                                                                                                                                                                                                                                                                                                                                                    call `num_occurs x (List.tl y)` from `num_occurs x y`
                                                                                                                                                                                                                                                                                                                                                                                                                                                    original:num_occurs x y
                                                                                                                                                                                                                                                                                                                                                                                                                                                    sub:num_occurs x (List.tl y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                    original ordinal:Ordinal.Int (_cnt y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                    sub ordinal:Ordinal.Int (_cnt (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                    path:[List.hd y = x && y <> []]
                                                                                                                                                                                                                                                                                                                                                                                                                                                    proof:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    detailed proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                    ground_instances:3
                                                                                                                                                                                                                                                                                                                                                                                                                                                    definitions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                    inductions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                    search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    0.008s
                                                                                                                                                                                                                                                                                                                                                                                                                                                    details:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                    smt_stats:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    num checks:8
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith assert lower:12
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith tableau max rows:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith tableau max columns:13
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith pivots:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                    rlimit count:4911
                                                                                                                                                                                                                                                                                                                                                                                                                                                    mk clause:26
                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype occurs check:22
                                                                                                                                                                                                                                                                                                                                                                                                                                                    mk bool var:89
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith assert upper:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype splits:3
                                                                                                                                                                                                                                                                                                                                                                                                                                                    decisions:27
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith row summations:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                    propagations:47
                                                                                                                                                                                                                                                                                                                                                                                                                                                    conflicts:11
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith fixed eqs:7
                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype accessor ax:14
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith conflicts:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith num rows:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                    datatype constructor ax:16
                                                                                                                                                                                                                                                                                                                                                                                                                                                    final checks:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                    added eqs:88
                                                                                                                                                                                                                                                                                                                                                                                                                                                    del clause:9
                                                                                                                                                                                                                                                                                                                                                                                                                                                    arith eq adapter:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                    memory:34.260000
                                                                                                                                                                                                                                                                                                                                                                                                                                                    max memory:44.370000
                                                                                                                                                                                                                                                                                                                                                                                                                                                    num allocs:53806273651.000000
                                                                                                                                                                                                                                                                                                                                                                                                                                                    Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • start[0.008s]
                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : int) = count.list (const 0) y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_1 : ty_0 list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_2 : int) = count.list (const 0) _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_3 : bool) = List.hd _x_1 = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_4 : bool) = _x_1 <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                        (List.hd y = x) && (y <> [] && ((_x_0 >= 0) && (_x_2 >= 0)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                        ==> (not (_x_3 && _x_4) && not (not _x_3 && _x_4))
                                                                                                                                                                                                                                                                                                                                                                                                                                                            || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                      into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : ty_0 list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_1 : bool) = List.hd _x_0 = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_2 : bool) = _x_0 <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_3 : int) = count.list (const 0) _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_4 : int) = count.list (const 0) y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                      (not (_x_1 && _x_2) && not (not _x_1 && _x_2))
                                                                                                                                                                                                                                                                                                                                                                                                                                                      || Ordinal.( << ) (Ordinal.Int _x_3) (Ordinal.Int _x_4)
                                                                                                                                                                                                                                                                                                                                                                                                                                                      || not ((List.hd y = x) && y <> [] && (_x_4 >= 0) && (_x_3 >= 0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      []
                                                                                                                                                                                                                                                                                                                                                                                                                                                      rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          (|Ordinal.<<| (|Ordinal.Int_79/boot|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (|count.list_3720/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|g…
                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                            (|count.list_3720/server| (|get.::.1_3699/server| y_3706/server))
                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              (|count.list_3720/server| y_3706/server)
                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Unsat
                                                                                                                                                                                                                                                                                                                                                                                                                                                              call `num_occurs x (List.tl y)` from `num_occurs x y`
                                                                                                                                                                                                                                                                                                                                                                                                                                                              original:num_occurs x y
                                                                                                                                                                                                                                                                                                                                                                                                                                                              sub:num_occurs x (List.tl y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                              original ordinal:Ordinal.Int (_cnt y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                              sub ordinal:Ordinal.Int (_cnt (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                              path:[not (List.hd y = x) && y <> []]
                                                                                                                                                                                                                                                                                                                                                                                                                                                              proof:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              detailed proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                              ground_instances:3
                                                                                                                                                                                                                                                                                                                                                                                                                                                              definitions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                              inductions:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                              search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              0.010s
                                                                                                                                                                                                                                                                                                                                                                                                                                                              details:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                              smt_stats:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              num checks:8
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith assert lower:12
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith tableau max rows:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith tableau max columns:13
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith pivots:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                              rlimit count:2466
                                                                                                                                                                                                                                                                                                                                                                                                                                                              mk clause:26
                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype occurs check:22
                                                                                                                                                                                                                                                                                                                                                                                                                                                              mk bool var:89
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith assert upper:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype splits:3
                                                                                                                                                                                                                                                                                                                                                                                                                                                              decisions:27
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith row summations:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                              propagations:47
                                                                                                                                                                                                                                                                                                                                                                                                                                                              conflicts:11
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith fixed eqs:7
                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype accessor ax:14
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith conflicts:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith num rows:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                              datatype constructor ax:16
                                                                                                                                                                                                                                                                                                                                                                                                                                                              final checks:6
                                                                                                                                                                                                                                                                                                                                                                                                                                                              added eqs:81
                                                                                                                                                                                                                                                                                                                                                                                                                                                              del clause:9
                                                                                                                                                                                                                                                                                                                                                                                                                                                              arith eq adapter:10
                                                                                                                                                                                                                                                                                                                                                                                                                                                              memory:34.210000
                                                                                                                                                                                                                                                                                                                                                                                                                                                              max memory:44.370000
                                                                                                                                                                                                                                                                                                                                                                                                                                                              num allocs:53370122493.000000
                                                                                                                                                                                                                                                                                                                                                                                                                                                              Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.010s]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : int) = count.list (const 0) y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_1 : ty_0 list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_2 : int) = count.list (const 0) _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_3 : bool) = List.hd _x_1 = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_4 : bool) = _x_1 <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  not (List.hd y = x) && (y <> [] && ((_x_0 >= 0) && (_x_2 >= 0)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  ==> (not (_x_3 && _x_4) && not (not _x_3 && _x_4))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_0 : ty_0 list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_1 : bool) = List.hd _x_0 = x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_2 : bool) = _x_0 <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_3 : int) = count.list (const 0) _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_4 : int) = count.list (const 0) y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                (not (_x_1 && _x_2) && not (not _x_1 && _x_2))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                || Ordinal.( << ) (Ordinal.Int _x_3) (Ordinal.Int _x_4)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                || not (not (List.hd y = x) && y <> [] && (_x_4 >= 0) && (_x_3 >= 0))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (|Ordinal.<<| (|Ordinal.Int_79/boot|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (|count.list_3720/server|
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|g…
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (|count.list_3720/server| (|get.::.1_3699/server| y_3706/server))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (|count.list_3720/server| y_3706/server)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • Unsat

                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Anticipating the need to reason about the interplay of num_occurs and merge, let's prove the following nice lemmas characterising their relationship (applying the max_induct 1 strategy to find these lemmas as above is a great exercise).

                                                                                                                                                                                                                                                                                                                                                                                                                                                                        In [26]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        theorem num_occur_merge a x y =
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         num_occurs a (merge x y) = num_occurs a x + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [@@induct functional merge]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [@@rewrite]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Out[26]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        val num_occur_merge : Z.t -> Z.t list -> Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        num_occurs a (merge x y) = num_occurs a x + num_occurs a y.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        We shall induct according to a scheme derived from merge.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (not ((List.hd x < List.hd y) && (y <> [] && x <> []))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          && not (not (List.hd x < List.hd y) && (y <> [] && x <> [])) ==> φ a x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         && ((x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              && (y <> [] && (not (List.hd x < List.hd y) && φ a x (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ==> φ a x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                             && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 && (y <> [] && ((List.hd x < List.hd y) && φ a (List.tl x) y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ==> φ a x y)).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        3 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 3:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. not (not (List.hd y <= List.hd x) && y <> [] && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H1. not ((List.hd y <= List.hd x) && y <> [] && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         num_occurs a (merge x y) = num_occurs a x + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definitions of merge and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H1. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H2. List.hd y <= List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H3. num_occurs a (merge x (List.tl y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = num_occurs a x + num_occurs a (List.tl y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         num_occurs a (merge x y) = num_occurs a x + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definitions of merge and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H1. y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H2. not (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         H3. num_occurs a (merge (List.tl x) y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = num_occurs a (List.tl x) + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         num_occurs a (merge x y) = num_occurs a x + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        But simplification reduces this to true, using the definitions of merge and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:def merge)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:def num_occurs)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (:induct merge)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        definitions:14
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        0.142s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • start[0.142s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            num_occurs ( :var_0: ) (merge ( :var_1: ) ( :var_2: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            = num_occurs ( :var_0: ) ( :var_1: ) + num_occurs ( :var_0: ) ( :var_2: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                          num_occurs a (merge x y) = num_occurs a x + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.142s, "1"] num_occurs a (merge x y) = num_occurs a x + num_occurs a y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • induction on (functional merge)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            :scheme (not ((List.hd x < List.hd y) && (y <> [] && x <> []))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     && not (not (List.hd x < List.hd y) && (y <> [] && x <> []))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ==> φ a x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    && ((x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         && (y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             && (not (List.hd x < List.hd y) && φ a x (List.tl y)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ==> φ a x y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            && (y <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && ((List.hd x < List.hd y) && φ a (List.tl x) y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            ==> φ a x y))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • Split (let (_x_0 : bool) = List.hd y <= List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_1 : bool) = not _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_2 : bool) = y <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_3 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_4 : int) = num_occurs a x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_5 : int) = num_occurs a y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_6 : bool) = num_occurs a (merge x y) = _x_4 + _x_5 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_7 : bool) = _x_3 && _x_2 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_8 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_9 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   (not (not (_x_1 && _x_2 && _x_3) && not (_x_0 && _x_2 && _x_3))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    || _x_6)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   && (not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (_x_7 && _x_0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        && (num_occurs a (merge x _x_8) = _x_4 + num_occurs a _x_8))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || _x_6)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   && (not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (_x_7 && _x_1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        && (num_occurs a (merge _x_9 y) = num_occurs a _x_9 + _x_5))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || _x_6)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   :cases [let (_x_0 : bool) = List.hd y <= List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : bool) = y <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (not _x_0 && _x_1 && _x_2) || (_x_0 && _x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || (num_occurs a (merge x y) = num_occurs a x + num_occurs a y);
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_0 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : int) = num_occurs a x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           not (x <> []) || not (y <> []) || not (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (num_occurs a (merge x _x_0) = _x_1 + num_occurs a _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || (num_occurs a (merge x y) = _x_1 + num_occurs a y);
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : int) = num_occurs a y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           not (x <> []) || not (y <> []) || (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (num_occurs a (merge _x_0 y) = num_occurs a _x_0 + _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           || (num_occurs a (merge x y) = num_occurs a x + _x_1)])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_0 : int list) = List.tl x in let (_x_1 : int) = num_occurs a y in not (x <> []) || not (y <> []) || (List.hd y <= List.hd x) || not (num_occurs a (merge _x_0 y) = num_occurs a _x_0 + _x_1) || (num_occurs a (merge x y) = num_occurs a x + _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.141s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_1 : int) = num_occurs a y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  not (x <> []) || not (y <> []) || (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  || not (num_occurs a (merge _x_0 y) = num_occurs a _x_0 + _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  || (num_occurs a (merge x y) = num_occurs a x + _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                [num_occurs, num_occurs, merge]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : int list) = List.tl y in let (_x_1 : int) = num_occurs a x in not (x <> []) || not (y <> []) || not (List.hd y <= List.hd x) || not (num_occurs a (merge x _x_0) = _x_1 + num_occurs a _x_0) || (num_occurs a (merge x y) = _x_1 + num_occurs a y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[0.141s, "2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : int list) = List.tl y in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_1 : int) = num_occurs a x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      not (x <> []) || not (y <> []) || not (List.hd y <= List.hd x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      || not (num_occurs a (merge x _x_0) = _x_1 + num_occurs a _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      || (num_occurs a (merge x y) = _x_1 + num_occurs a y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    [num_occurs, num_occurs, merge]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : bool) = List.hd y <= List.hd x in let (_x_1 : bool) = y <> [] in let (_x_2 : bool) = x <> [] in (not _x_0 && _x_1 && _x_2) || (_x_0 && _x_1 && _x_2) || (num_occurs a (merge x y) = num_occurs a x + num_occurs a y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.141s, "3"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : bool) = List.hd y <= List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_1 : bool) = y <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (not _x_0 && _x_1 && _x_2) || (_x_0 && _x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          || (num_occurs a (merge x y) = num_occurs a x + num_occurs a y)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [num_occurs, merge, num_occurs, merge, num_occurs, merge, num_occurs, merge]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    In [27]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    theorem num_occurs_arith (x : int list) (a : int) =
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      x <> [] && List.tl x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ==>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      =
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      num_occurs a x - num_occurs a (odds x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      [@@induct functional num_occurs]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      [@@rewrite]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Out[27]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    val num_occurs_arith : Z.t list -> Z.t -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (x <> []) && (List.tl x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ==> num_occurs a (odds (List.tl x)) = num_occurs a x - num_occurs a (odds x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    We shall induct according to a scheme derived from num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (not (not (List.hd x = a) && x <> []) && not ((List.hd x = a) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     && ((x <> [] && ((List.hd x = a) && φ a (List.tl x)) ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         && (x <> [] && (not (List.hd x = a) && φ a (List.tl x)) ==> φ a x)).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    3 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 3:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. not (x = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. not (List.tl x = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. not (not (List.hd x = a) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. not ((List.hd x = a) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs a x + (-1) * num_occurs a (odds x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. not (x = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. not (List.tl x = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H4. not (List.tl x = []) && not (List.tl (List.tl x) = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ==> num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = num_occurs a (List.tl x) + (-1) * num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs a x + (-1) * num_occurs a (odds x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of num_occurs and odds to the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    following 4 subgoals:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2.4:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl (List.tl x)) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. num_occurs (List.hd x) (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs (List.hd x) (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs (List.hd x) (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. num_occurs (List.hd x) (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs (List.hd x) (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs (List.hd x) (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. List.hd (List.tl x) = List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2.3:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl (List.tl x)) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. num_occurs (List.hd x) (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = 1 + num_occurs (List.hd x) (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs (List.hd x) (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H4. List.hd (List.tl x) = List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     num_occurs (List.hd x) (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs (List.hd x) (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       + (-1) * num_occurs (List.hd x) (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs (List.hd x) (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs (List.hd x) (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. List.hd (List.tl x) = List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of num_occurs and odds to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2.2':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. 0 = num_occurs (List.hd x) (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. List.hd (List.tl x) = List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. List.hd (List.tl x) = List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs (List.hd x) (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       + (-1) * num_occurs (List.hd x) (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of num_occurs and odds to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 2.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. List.hd (List.tl x) = List.hd x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     0 = num_occurs (List.hd x) (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. not (x = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. not (List.tl x = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. not (List.hd x = a)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H4. not (List.tl x = []) && not (List.tl (List.tl x) = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ==> num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = num_occurs a (List.tl x) + (-1) * num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs a x + (-1) * num_occurs a (odds x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of num_occurs and odds to the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    following 4 subgoals:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.4:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl (List.tl x)) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs a (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs a (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C2. List.hd (List.tl x) = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.3:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl (List.tl x)) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = 1 + num_occurs a (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H4. List.hd (List.tl x) = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs a (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs a (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C2. List.hd (List.tl x) = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of num_occurs and odds to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.2':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. 0 = num_occurs a (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C2. List.hd (List.tl x) = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H3. List.hd (List.tl x) = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         = num_occurs a (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           + (-1) * num_occurs a (odds (List.tl (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    This simplifies, using the definitions of num_occurs and odds to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Subgoal 1.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H0. List.tl (List.tl x) = []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H2. (List.tl x) <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C0. 0 = num_occurs (List.hd (List.tl x)) (List.tl (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     C1. List.hd x = List.hd (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:def num_occurs)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:def odds)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (:induct num_occurs)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    definitions:48
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    0.828s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • start[0.828s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : int list) = List.tl ( :var_0: ) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        not (( :var_0: ) = []) && not (_x_0 = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        ==> num_occurs ( :var_1: ) (odds _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            = num_occurs ( :var_1: ) ( :var_0: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              - num_occurs ( :var_1: ) (odds ( :var_0: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : int list) = List.tl x in (x = []) || (_x_0 = []) || (num_occurs a (odds _x_0) = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[0.828s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (x = []) || (_x_0 = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          || (num_occurs a (odds _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • induction on (functional num_occurs)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        :scheme (not (not (List.hd x = a) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 && not ((List.hd x = a) && x <> []) ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && ((x <> [] && ((List.hd x = a) && φ a (List.tl x)) ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    && (x <> [] && (not (List.hd x = a) && φ a (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        ==> φ a x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • Split (let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_1 : bool) = not (_x_0 = []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_2 : int) = num_occurs a (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_3 : bool)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   = not (not (x = []) && _x_1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     || (_x_2 = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_4 : bool) = List.hd x = a in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_5 : bool) = not _x_4 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_6 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_7 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               let (_x_8 : bool)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   = not (_x_1 && not (_x_7 = []))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     || (num_occurs a (odds _x_7) = num_occurs a _x_0 + (-1) * _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (_x_3 || not (not (_x_5 && _x_6) && not (_x_4 && _x_6)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               && (_x_3 || not (_x_6 && _x_4 && _x_8))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               && (_x_3 || not (_x_6 && _x_5 && _x_8))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               :cases [let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_1 : bool) = List.hd x = a in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (x = []) || (_x_0 = []) || (not _x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || (_x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || (num_occurs a (odds _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           = num_occurs a x + (-1) * num_occurs a (odds x));
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_1 : bool) = _x_0 = [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_2 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_3 : int) = num_occurs a (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (x = []) || _x_1 || not (x <> []) || not (List.hd x = a)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (not _x_1 && not (_x_2 = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ==> num_occurs a (odds _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               = num_occurs a _x_0 + (-1) * _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || (_x_3 = num_occurs a x + (-1) * num_occurs a (odds x));
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_1 : bool) = _x_0 = [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_2 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       let (_x_3 : int) = num_occurs a (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (x = []) || _x_1 || not (x <> []) || (List.hd x = a)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (not _x_1 && not (_x_2 = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ==> num_occurs a (odds _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               = num_occurs a _x_0 + (-1) * _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       || (_x_3 = num_occurs a x + (-1) * num_occurs a (odds x))])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : int list) = List.tl x in let (_x_1 : bool) = _x_0 = [] in let (_x_2 : int list) = List.tl _x_0 in let (_x_3 : int) = num_occurs a (odds _x_0) in (x = []) || _x_1 || not (x <> []) || (List.hd x = a) || not (not _x_1 && not (_x_2 = []) ==> num_occurs a (odds _x_2) = num_occurs a _x_0 + (-1) * _x_3) || (_x_3 = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • start[0.827s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_1 : bool) = _x_0 = [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_2 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_3 : int) = num_occurs a (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (x = []) || _x_1 || not (x <> []) || (List.hd x = a)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not _x_1 && not (_x_2 = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  ==> num_occurs a (odds _x_2) = num_occurs a _x_0 + (-1) * _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              || (_x_3 = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_1 : bool) = (List.hd x = a) || (x = []) || (_x_0 = []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_2 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_3 : bool) = _x_2 = [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_4 : bool) = _x_1 || _x_3 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_5 : int) = num_occurs a (odds _x_2) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_6 : int) = num_occurs a _x_2 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_7 : int) = num_occurs a (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_8 : int) = (-1) * _x_7 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_9 : int) = num_occurs a _x_0 + (-1) * _x_5 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_10 : bool) = _x_7 = _x_9 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_11 : bool) = List.hd _x_0 = a in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_12 : bool) = not _x_11 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            let (_x_13 : bool) = _x_1 || not _x_3 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (_x_4 || not (_x_5 = _x_6 + _x_8) || _x_10 || _x_11)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            && (_x_4 || _x_10 || not (_x_5 = 1 + _x_6 + _x_8) || _x_12)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            && (_x_13 || (0 = _x_9) || _x_11) && (_x_13 || (1 = _x_9) || _x_12)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            [num_occurs, odds, num_occurs, num_occurs, num_occurs, num_occurs, odds,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             num_occurs, odds, num_occurs, num_occurs, odds, num_occurs, num_occurs]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              let (_x_0 : int list) = List.tl x in let (_x_1 : bool) = _x_0 = [] in let (_x_2 : int list) = List.tl _x_0 in let (_x_3 : int) = num_occurs a (odds _x_0) in (x = []) || _x_1 || not (x <> []) || not (List.hd x = a) || not (not _x_1 && not (_x_2 = []) ==> num_occurs a (odds _x_2) = num_occurs a _x_0 + (-1) * _x_3) || (_x_3 = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • start[0.827s, "2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_1 : bool) = _x_0 = [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_2 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_3 : int) = num_occurs a (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (x = []) || _x_1 || not (x <> []) || not (List.hd x = a)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  || not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (not _x_1 && not (_x_2 = [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ==> num_occurs a (odds _x_2) = num_occurs a _x_0 + (-1) * _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  || (_x_3 = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_1 : bool) = (x = []) || (_x_0 = []) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_2 : int list) = List.tl _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_3 : bool) = _x_2 = [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_4 : bool) = _x_1 || _x_3 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_5 : int) = List.hd x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_6 : int) = num_occurs _x_5 (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_7 : int) = num_occurs _x_5 (odds _x_2) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_8 : int) = num_occurs _x_5 _x_0 + (-1) * _x_7 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_9 : bool) = _x_6 = _x_8 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_10 : int) = num_occurs _x_5 _x_2 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_11 : int) = (-1) * _x_6 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_12 : bool) = List.hd _x_0 = _x_5 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_13 : bool) = not _x_12 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                let (_x_14 : bool) = _x_1 || not _x_3 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (_x_4 || _x_9 || not (_x_7 = _x_10 + _x_11) || _x_12)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && (_x_4 || not (_x_7 = 1 + _x_10 + _x_11) || _x_13 || _x_9)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && (_x_14 || (0 = _x_8) || _x_12) && (_x_14 || (1 = _x_8) || _x_13)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                [num_occurs, odds, num_occurs, num_occurs, num_occurs, num_occurs, odds,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 num_occurs, odds, num_occurs, num_occurs, odds, num_occurs, num_occurs]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  let (_x_0 : int list) = List.tl x in let (_x_1 : bool) = List.hd x = a in let (_x_2 : bool) = x <> [] in (x = []) || (_x_0 = []) || (not _x_1 && _x_2) || (_x_1 && _x_2) || (num_occurs a (odds _x_0) = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[0.827s, "3"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_1 : bool) = List.hd x = a in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (x = []) || (_x_0 = []) || (not _x_1 && _x_2) || (_x_1 && _x_2)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      || (num_occurs a (odds _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          = num_occurs a x + (-1) * num_occurs a (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    true
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      forward_chaining:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Finally, let's put the pieces together and prove our main remaining theorem.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                And now, our second main theorem at last:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                In [28]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                theorem merge_sort_elements a (x : int list) =
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 num_occurs a x = num_occurs a (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                [@@induct functional merge_sort]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Out[28]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                val merge_sort_elements : Z.t -> Z.t list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Goal:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                num_occurs a x = num_occurs a (merge_sort x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1 nontautological subgoal.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We shall induct according to a scheme derived from merge_sort.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Induction scheme:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (not (not Is_a([], List.tl x) && x <> []) ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     && (not Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         && (φ a (odds (List.tl x)) && φ a (odds x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ==> φ a x).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2 nontautological subgoals.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. not Is_a([], List.tl x) && x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. num_occurs a x = num_occurs a (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definitions of merge_sort and num_occurs to the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                following 3 subgoals:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 2.3:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. num_occurs a x = 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 2.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. num_occurs a x = 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definition of num_occurs to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 2.2':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. num_occurs a (List.tl x) = 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 2.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H2. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 num_occurs a x = 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definition of num_occurs to:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 2.1':
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 num_occurs (List.hd x) (List.tl x) = 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the definition of num_occurs.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. not Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H2. num_occurs a (odds (List.tl x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs a (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H3. num_occurs a (odds x) = num_occurs a (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 num_occurs a x = num_occurs a (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                This simplifies, using the definitions of merge_sort and num_occurs, and the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rewrite rule num_occurs_arith to the following 2 subgoals:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.2:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. num_occurs a (odds x) = num_occurs a (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H2. num_occurs a (List.tl x) + (-1) * num_occurs a (odds x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs a (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C2. num_occurs a (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = num_occurs a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (merge (merge_sort (odds x)) (merge_sort (odds (List.tl x))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the rewrite rule
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                num_occur_merge.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Subgoal 1.1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H0. num_occurs a (odds x) = num_occurs a (merge_sort (odds x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H1. x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H2. List.hd x = a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 H3. num_occurs a (List.tl x) + (-1) * num_occurs a (odds x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = (-1) + num_occurs a (merge_sort (odds (List.tl x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |---------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C0. Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C1. num_occurs a (List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     = (-1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       + num_occurs a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (merge (merge_sort (odds x)) (merge_sort (odds (List.tl x))))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                But simplification reduces this to true, using the rewrite rule
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                num_occur_merge.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Rules:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:def num_occurs)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:rw num_occur_merge)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:rw num_occurs_arith)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (:induct merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Proved
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                proof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ground_instances:0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                definitions:17
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                inductions:1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                search_time:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1.272s
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Expand
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • start[1.272s, "Goal"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    num_occurs ( :var_0: ) ( :var_1: )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    = num_occurs ( :var_0: ) (merge_sort ( :var_1: ))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  num_occurs a x = num_occurs a (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • start[1.272s, "1"] num_occurs a x = num_occurs a (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • induction on (functional merge_sort)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    :scheme (not (not Is_a([], List.tl x) && x <> []) ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            && (x <> []
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && (not Is_a([], List.tl x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    && (φ a (odds (List.tl x)) && φ a (odds x)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ==> φ a x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • Split (let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_1 : bool) = not Is_a([], _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_2 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_3 : bool) = num_occurs a x = num_occurs a (merge_sort x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_4 : int list) = odds _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           let (_x_5 : int list) = odds x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ((_x_1 && _x_2) || _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           && (not
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (_x_2 && _x_1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && (num_occurs a _x_4 = num_occurs a (merge_sort _x_4))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                && (num_occurs a _x_5 = num_occurs a (merge_sort _x_5)))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               || _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           :cases [(not Is_a([], List.tl x) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || (num_occurs a x = num_occurs a (merge_sort x));
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_1 : int list) = odds _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   let (_x_2 : int list) = odds x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   not (x <> []) || Is_a([], _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || not (num_occurs a _x_1 = num_occurs a (merge_sort _x_1))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || not (num_occurs a _x_2 = num_occurs a (merge_sort _x_2))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   || (num_occurs a x = num_occurs a (merge_sort x))])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      let (_x_0 : int list) = List.tl x in let (_x_1 : int list) = odds _x_0 in let (_x_2 : int list) = odds x in not (x <> []) || Is_a([], _x_0) || not (num_occurs a _x_1 = num_occurs a (merge_sort _x_1)) || not (num_occurs a _x_2 = num_occurs a (merge_sort _x_2)) || (num_occurs a x = num_occurs a (merge_sort x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • start[1.272s, "1"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_1 : int list) = odds _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_2 : int list) = odds x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          not (x <> []) || Is_a([], _x_0)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          || not (num_occurs a _x_1 = num_occurs a (merge_sort _x_1))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          || not (num_occurs a _x_2 = num_occurs a (merge_sort _x_2))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          || (num_occurs a x = num_occurs a (merge_sort x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_0 : int list) = List.tl x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_1 : int list) = odds x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_2 : int) = num_occurs a _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_3 : int list) = merge_sort _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_4 : bool)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            = Is_a([], _x_0) || not (_x_2 = num_occurs a _x_3) || not (x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_5 : bool) = List.hd x = a in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_6 : int) = num_occurs a _x_0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_7 : int) = _x_6 + (-1) * _x_2 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_8 : int list) = merge_sort (odds _x_0) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_9 : int) = num_occurs a _x_8 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        let (_x_10 : int) = num_occurs a (merge _x_3 _x_8) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (_x_4 || _x_5 || not (_x_7 = _x_9) || (_x_6 = _x_10))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        && (_x_4 || not _x_5 || (_x_6 = (-1) + _x_10) || not (_x_7 = (-1) + _x_9))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [merge_sort, num_occurs, merge_sort, num_occurs, num_occurs]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        num_occurs_arith
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (not Is_a([], List.tl x) && x <> []) || (num_occurs a x = num_occurs a (merge_sort x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • start[1.272s, "2"]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (not Is_a([], List.tl x) && x <> [])
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            || (num_occurs a x = num_occurs a (merge_sort x))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • simplify
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          into:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_0 : bool) = not Is_a([], List.tl x) in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_1 : bool) = x <> [] in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_2 : int) = num_occurs a x in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_3 : bool) = _x_2 = 0 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_4 : bool) = _x_0 || not _x_1 in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          let (_x_5 : bool) = List.hd x = a in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (_x_0 || _x_1 || _x_3) && (_x_4 || _x_5 || _x_3)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          && (_x_4 || not _x_5 || (_x_2 = 1))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          [num_occurs, merge_sort, num_occurs, num_occurs, num_occurs, num_occurs,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           merge_sort]
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          rewrite_steps:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            forward_chaining:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Subproof
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • Subproof

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      So now we're sure merge_sort really is a proper sorting function.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Merge Sort is Correct!

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      To recap, we've proved:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      theorem merge_sort_sorts x =
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       is_sorted (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      and

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      theorem merge_sort_elements a (x : int list) =
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       num_occurs a x = num_occurs a (merge_sort x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Beautiful! Happy proving (and sorting)!