Unrolling

The first tool that Imandra makes available in our verification toolbox is recursive function unrolling, a form of bounded model checking backed by Satisfiability Modulo Theories (SMT) solving. This technique is completely automatic, and is in general not influenced by the presence of proved rules or enabled/disabled functions (except with used in conjunction with the [@@simp] attribute).

Completeness

For many classes of problems, unrolling is "complete" in various senses. For example, for goals involving only non-recursive functions, algebraic datatypes and linear arithmetic, unrolling will always be able to prove a true goal or refute a false goal in a finite amount of time and space. Moreover, for an even wider class of problems involving recursive functions, datatypes and arithmetic, unrolling is "complete for counterexamples." This means that if a counterexample to a goal exists, unrolling will in principle always be able to synthesize one. This relies on Imandra's "fair" strategy for incrementally expanding the "interpretation call-graph" of a goal.

That said, part of the beauty of unrolling is that you don't need to understand it to apply it!

Strategy

In general, it's recommended to apply unrolling to a goal before you attempt other methods such as the inductive waterfall ([@@auto]). It's amazing how often seemingly true goals are false due to subtle edge cases, and the ability of unrolling to construct concrete counterexamples can be an invaluable filter on your conjectures.

Examples

To use unrolling, we simply use the verify or instance commands.

Let's use unrolling to find an instance of two lists of integers, whose sum equals the length of the two lists concatenated. We shall constrain the total length of the two lists to be positive (for fun, at least 10), so we obtain something more interesting than the simple x=[],y=[] solution!

In [1]:
instance
  (fun x y -> List.length (x@y) > 10
    && List.fold_left (+) 0 (x@y) = List.length (x@y))
Out[1]:
- : Z.t list -> Z.t list -> bool = <fun>
module CX : sig val x : Z.t list val y : Z.t list end
Instance (after 29 steps, 0.033s):
let x : int list = []
let y : int list =
  [590; 7719; (-16563); 8365; 41; 5920; 974; 1323; 2240; 9725; (-20323)]
Instance
proof attempt
ground_instances:29
definitions:0
inductions:0
search_time:
0.033s
details:
Expand
smt_stats:
num checks:59
arith assert lower:119
arith tableau max rows:36
arith tableau max columns:64
arith pivots:48
rlimit count:30300
mk clause:195
datatype occurs check:231
mk bool var:982
arith assert upper:83
datatype splits:48
decisions:547
arith row summations:389
propagations:567
conflicts:64
arith fixed eqs:64
datatype accessor ax:70
minimized lits:3
arith conflicts:22
arith num rows:36
arith assert diseq:39
datatype constructor ax:225
num allocs:973284
final checks:59
added eqs:1788
del clause:165
arith eq adapter:123
memory:5.850000
max memory:5.850000
Expand
  • start[0.033s]
      let (_x_0 : int list) = List.append ( :var_0: ) ( :var_1: ) in
      let (_x_1 : int) = List.length _x_0 in
      _x_1 > 10 && List.fold_left ( + ) 0 _x_0 = _x_1
  • simplify

    into:
    let (_x_0 : int list) = List.append ( :var_0: ) ( :var_1: ) in
    let (_x_1 : int) = List.length _x_0 in
    not (_x_1 <= 10) && List.fold_left ( + ) 0 _x_0 = _x_1
    expansions:
    []
    rewrite_steps:
      forward_chaining:
      • unroll
        expr:
        (|List.fold_left_1635/server|
          0
          (|List.append_1626/server| x_1345/client y_1346/client))
        expansions:
        • unroll
          expr:
          (|List.length_1631/server|
            (|List.append_1626/server| x_1345/client y_1346/client))
          expansions:
          • unroll
            expr:
            (|List.append_1626/server| x_1345/client y_1346/client)
            expansions:
            • unroll
              expr:
              (|List.fold_left_1635/server|
                (|get.::.0_1624/server|
                  (|List.append_1626/server| x_1345/client…
              expansions:
              • unroll
                expr:
                (|List.length_1631/server|
                  (|get.::.1_1625/server|
                    (|List.append_1626/server| x_1345/client y_…
                expansions:
                • unroll
                  expr:
                  (let ((a!1 (+ (|get.::.0_1624/server|
                                  (|List.append_1626/server| x_1345/client y_134…
                  expansions:
                  • unroll
                    expr:
                    (|List.length_1631/server|
                      (|get.::.1_1625/server|
                        (|get.::.1_1625/server|
                          (|List.append…
                    expansions:
                    • unroll
                      expr:
                      (let ((a!1 (|get.::.0_1624/server|
                                   (|get.::.1_1625/server|
                                     (|get.::.1_16…
                      expansions:
                      • unroll
                        expr:
                        (let ((a!1 (|get.::.1_1625/server|
                                     (|get.::.1_1625/server|
                                       (|get.::.1_16…
                        expansions:
                        • unroll
                          expr:
                          (|List.append_1626/server| (|get.::.1_1625/server| x_1345/client) y_1346/client)
                          expansions:
                          • unroll
                            expr:
                            (let ((a!1 (|get.::.0_1624/server|
                                         (|get.::.1_1625/server|
                                           (|get.::.1_16…
                            expansions:
                            • unroll
                              expr:
                              (let ((a!1 (|get.::.1_1625/server|
                                           (|get.::.1_1625/server|
                                             (|get.::.1_16…
                              expansions:
                              • unroll
                                expr:
                                (let ((a!1 (|get.::.0_1624/server|
                                             (|get.::.1_1625/server|
                                               (|get.::.1_16…
                                expansions:
                                • unroll
                                  expr:
                                  (let ((a!1 (|get.::.1_1625/server|
                                               (|get.::.1_1625/server|
                                                 (|get.::.1_16…
                                  expansions:
                                  • unroll
                                    expr:
                                    (|List.append_1626/server|
                                      (|get.::.1_1625/server| (|get.::.1_1625/server| x_1345/client))
                                      y_134…
                                    expansions:
                                    • unroll
                                      expr:
                                      (let ((a!1 (|get.::.0_1624/server|
                                                   (|get.::.1_1625/server|
                                                     (|get.::.1_16…
                                      expansions:
                                      • unroll
                                        expr:
                                        (let ((a!1 (|get.::.1_1625/server|
                                                     (|get.::.1_1625/server|
                                                       (|get.::.1_16…
                                        expansions:
                                        • unroll
                                          expr:
                                          (let ((a!1 (|get.::.0_1624/server|
                                                       (|get.::.1_1625/server|
                                                         (|get.::.1_16…
                                          expansions:
                                          • unroll
                                            expr:
                                            (let ((a!1 (|get.::.1_1625/server|
                                                         (|get.::.1_1625/server|
                                                           (|get.::.1_16…
                                            expansions:
                                            • unroll
                                              expr:
                                              (|List.append_1626/server|
                                                (|get.::.1_1625/server|
                                                  (|get.::.1_1625/server| (|get.::.1_1625/ser…
                                              expansions:
                                              • unroll
                                                expr:
                                                (let ((a!1 (|get.::.0_1624/server|
                                                             (|get.::.1_1625/server|
                                                               (|get.::.1_16…
                                                expansions:
                                                • unroll
                                                  expr:
                                                  (let ((a!1 (|get.::.1_1625/server|
                                                               (|get.::.1_1625/server|
                                                                 (|get.::.1_16…
                                                  expansions:
                                                  • unroll
                                                    expr:
                                                    (let ((a!1 (|get.::.0_1624/server|
                                                                 (|get.::.1_1625/server|
                                                                   (|get.::.1_16…
                                                    expansions:
                                                    • unroll
                                                      expr:
                                                      (let ((a!1 (|get.::.1_1625/server|
                                                                   (|get.::.1_1625/server|
                                                                     (|get.::.1_16…
                                                      expansions:
                                                      • unroll
                                                        expr:
                                                        (let ((a!1 (|get.::.1_1625/server|
                                                                     (|get.::.1_1625/server|
                                                                       (|get.::.1_16…
                                                        expansions:
                                                        • unroll
                                                          expr:
                                                          (let ((a!1 (|get.::.0_1624/server|
                                                                       (|get.::.1_1625/server|
                                                                         (|get.::.1_16…
                                                          expansions:
                                                          • unroll
                                                            expr:
                                                            (let ((a!1 (|get.::.1_1625/server|
                                                                         (|get.::.1_1625/server|
                                                                           (|get.::.1_16…
                                                            expansions:
                                                            • unroll
                                                              expr:
                                                              (let ((a!1 (|get.::.0_1624/server|
                                                                           (|get.::.1_1625/server|
                                                                             (|get.::.1_16…
                                                              expansions:
                                                              • unroll
                                                                expr:
                                                                (let ((a!1 (|get.::.1_1625/server|
                                                                             (|get.::.1_1625/server|
                                                                               (|get.::.1_16…
                                                                expansions:
                                                                • Sat (Some let x : int list = [] let y : int list = [(Z.of_nativeint (590n)); (Z.of_nativeint (7719n)); (Z.of_nativeint (-16563n)); (Z.of_nativeint (8365n)); (Z.of_nativeint (41n)); (Z.of_nativeint (5920n)); (Z.of_nativeint (974n)); (Z.of_nativeint (1323n)); (Z.of_nativeint (2240n)); (Z.of_nativeint (9725n)); (Z.of_nativeint (-20323n))] )

                                                                Imandra was able to find a solution instantly, and reflected it into our runtime in the CX module. Let's compute with it to better understand it:

                                                                In [2]:
                                                                List.length (CX.x@CX.y);;
                                                                List.fold_left (+) 0 (CX.x@CX.y);;
                                                                
                                                                Out[2]:
                                                                - : Z.t = 11
                                                                - : Z.t = 11
                                                                

                                                                Unrolling limits

                                                                Unrolling works by creating a symbolic call graph for the negation of the goal we've asked Imandra to verify (or dually the positive version of the goal in the case of instance), and by iteratively extending this graph with incremental interpretations of recursive functions, up to a given unrolling bound, checking at each step for satisfiability.

                                                                The unrolling bound defaults to 100 and can be controlled globally using the #unroll <n> directive, or local to a given verify or instance call using the ~upto:<n> parameter.

                                                                If at any step of the unrolling process the negation of our original goal is satisfiable w.r.t. the interpreted approximations of the recursive functions, then Imandra has found a counterexample for our original goal, which has thus been refuted. In this case, Imandra will report Counterexample (after m steps) and install the found counterexample in the CX module.

                                                                If, on the other hand, Imandra is able to prove that there is no counterexample in a manner that is independent of the bound on the approximations, then our original goal is indeed a theorem valid for all possible inputs, and Imandra will report Theorem Proved. This can always be done for a wide class of theorems on catamorphisms (e.g., List.fold_right), for example.

                                                                Otherwise, if Imandra failed to find a counterexample or proof and stopped unrolling at the unrolling bound, we obtain a weaker result of the form Unknown (verified up to depth k), which effectively means: this may or may not be a theorem, but there are no counterexamples up to depth k. Such bounded results can nevertheless be very useful.

                                                                Let's try to understand in practice how the unrolling bound plays into unrolling. Consider this simple function that recursively decreases an integer until it reaches 0, then returns 1:

                                                                In [3]:
                                                                let rec f x =
                                                                  if x <= 0 then
                                                                    1
                                                                  else
                                                                    f (x - 1)
                                                                
                                                                Out[3]:
                                                                val f : Z.t -> Z.t = <fun>
                                                                
                                                                termination proof

                                                                Termination proof

                                                                call `f (x - 1)` from `f x`
                                                                original:f x
                                                                sub:f (x - 1)
                                                                original ordinal:Ordinal.Int (_cnt x)
                                                                sub ordinal:Ordinal.Int (_cnt (x - 1))
                                                                path:[not (x <= 0)]
                                                                proof:
                                                                detailed proof
                                                                ground_instances:1
                                                                definitions:0
                                                                inductions:0
                                                                search_time:
                                                                0.009s
                                                                details:
                                                                Expand
                                                                smt_stats:
                                                                num checks:3
                                                                arith assert lower:7
                                                                arith tableau max rows:4
                                                                arith tableau max columns:9
                                                                arith pivots:2
                                                                rlimit count:31327
                                                                mk clause:4
                                                                datatype occurs check:2
                                                                mk bool var:17
                                                                arith assert upper:2
                                                                decisions:2
                                                                arith row summations:3
                                                                propagations:2
                                                                conflicts:2
                                                                arith fixed eqs:2
                                                                datatype accessor ax:2
                                                                arith conflicts:1
                                                                arith num rows:4
                                                                num allocs:5286261
                                                                final checks:1
                                                                added eqs:4
                                                                del clause:4
                                                                arith eq adapter:2
                                                                memory:5.740000
                                                                max memory:5.850000
                                                                Expand
                                                                • start[0.009s]
                                                                    let (_x_0 : int) = if x >= 0 then x else 0 in
                                                                    let (_x_1 : int) = x - 1 in
                                                                    let (_x_2 : int) = if _x_1 >= 0 then _x_1 else 0 in
                                                                    not (x <= 0) && _x_0 >= 0 && _x_2 >= 0
                                                                    ==> _x_1 <= 0 || Ordinal.( << ) (Ordinal.Int _x_2) (Ordinal.Int _x_0)
                                                                • simplify
                                                                  into:
                                                                  (Ordinal.( << ) (Ordinal.Int (if x >= 1 then (-1) + x else 0))
                                                                   (Ordinal.Int (if x >= 0 then x else 0)) || x <= 1)
                                                                  || x <= 0
                                                                  expansions:
                                                                  []
                                                                  rewrite_steps:
                                                                    forward_chaining:
                                                                    • unroll
                                                                      expr:
                                                                      (|Ordinal.<<_129/client|
                                                                        (|Ordinal.Int_114/client|
                                                                          (ite (>= x_1905/server 1) (+ (- 1) x_1905/s…
                                                                      expansions:
                                                                      • Unsat

                                                                      Let's verify that for all x < 100, the function will return 1:

                                                                      In [4]:
                                                                      verify (fun x -> x < 100 ==> f x = 1)
                                                                      
                                                                      Out[4]:
                                                                      - : Z.t -> bool = <fun>
                                                                      
                                                                      Proved
                                                                      proof
                                                                      ground_instances:100
                                                                      definitions:0
                                                                      inductions:0
                                                                      search_time:
                                                                      0.159s
                                                                      details:
                                                                      Expand
                                                                      smt_stats:
                                                                      num checks:201
                                                                      arith assert lower:99
                                                                      arith tableau max rows:100
                                                                      arith tableau max columns:202
                                                                      rlimit count:28985
                                                                      mk clause:300
                                                                      mk bool var:503
                                                                      arith assert upper:1
                                                                      propagations:297
                                                                      conflicts:101
                                                                      arith fixed eqs:2
                                                                      arith num rows:100
                                                                      num allocs:14748865
                                                                      final checks:100
                                                                      added eqs:300
                                                                      del clause:297
                                                                      time:0.002000
                                                                      memory:7.810000
                                                                      max memory:7.810000
                                                                      Expand
                                                                      • start[0.159s] ( :var_0: ) < 100 ==> f ( :var_0: ) = 1
                                                                      • simplify

                                                                        into:
                                                                        100 <= ( :var_0: ) || f ( :var_0: ) = 1
                                                                        expansions:
                                                                        []
                                                                        rewrite_steps:
                                                                          forward_chaining:
                                                                          • unroll
                                                                            expr:
                                                                            (f_1348/client x_1350/client)
                                                                            expansions:
                                                                            • unroll
                                                                              expr:
                                                                              (f_1348/client (+ (- 1) x_1350/client))
                                                                              expansions:
                                                                              • unroll
                                                                                expr:
                                                                                (f_1348/client (+ (- 2) x_1350/client))
                                                                                expansions:
                                                                                • unroll
                                                                                  expr:
                                                                                  (f_1348/client (+ (- 3) x_1350/client))
                                                                                  expansions:
                                                                                  • unroll
                                                                                    expr:
                                                                                    (f_1348/client (+ (- 4) x_1350/client))
                                                                                    expansions:
                                                                                    • unroll
                                                                                      expr:
                                                                                      (f_1348/client (+ (- 5) x_1350/client))
                                                                                      expansions:
                                                                                      • unroll
                                                                                        expr:
                                                                                        (f_1348/client (+ (- 6) x_1350/client))
                                                                                        expansions:
                                                                                        • unroll
                                                                                          expr:
                                                                                          (f_1348/client (+ (- 7) x_1350/client))
                                                                                          expansions:
                                                                                          • unroll
                                                                                            expr:
                                                                                            (f_1348/client (+ (- 8) x_1350/client))
                                                                                            expansions:
                                                                                            • unroll
                                                                                              expr:
                                                                                              (f_1348/client (+ (- 9) x_1350/client))
                                                                                              expansions:
                                                                                              • unroll
                                                                                                expr:
                                                                                                (f_1348/client (+ (- 10) x_1350/client))
                                                                                                expansions:
                                                                                                • unroll
                                                                                                  expr:
                                                                                                  (f_1348/client (+ (- 11) x_1350/client))
                                                                                                  expansions:
                                                                                                  • unroll
                                                                                                    expr:
                                                                                                    (f_1348/client (+ (- 12) x_1350/client))
                                                                                                    expansions:
                                                                                                    • unroll
                                                                                                      expr:
                                                                                                      (f_1348/client (+ (- 13) x_1350/client))
                                                                                                      expansions:
                                                                                                      • unroll
                                                                                                        expr:
                                                                                                        (f_1348/client (+ (- 14) x_1350/client))
                                                                                                        expansions:
                                                                                                        • unroll
                                                                                                          expr:
                                                                                                          (f_1348/client (+ (- 15) x_1350/client))
                                                                                                          expansions:
                                                                                                          • unroll
                                                                                                            expr:
                                                                                                            (f_1348/client (+ (- 16) x_1350/client))
                                                                                                            expansions:
                                                                                                            • unroll
                                                                                                              expr:
                                                                                                              (f_1348/client (+ (- 17) x_1350/client))
                                                                                                              expansions:
                                                                                                              • unroll
                                                                                                                expr:
                                                                                                                (f_1348/client (+ (- 18) x_1350/client))
                                                                                                                expansions:
                                                                                                                • unroll
                                                                                                                  expr:
                                                                                                                  (f_1348/client (+ (- 19) x_1350/client))
                                                                                                                  expansions:
                                                                                                                  • unroll
                                                                                                                    expr:
                                                                                                                    (f_1348/client (+ (- 20) x_1350/client))
                                                                                                                    expansions:
                                                                                                                    • unroll
                                                                                                                      expr:
                                                                                                                      (f_1348/client (+ (- 21) x_1350/client))
                                                                                                                      expansions:
                                                                                                                      • unroll
                                                                                                                        expr:
                                                                                                                        (f_1348/client (+ (- 22) x_1350/client))
                                                                                                                        expansions:
                                                                                                                        • unroll
                                                                                                                          expr:
                                                                                                                          (f_1348/client (+ (- 23) x_1350/client))
                                                                                                                          expansions:
                                                                                                                          • unroll
                                                                                                                            expr:
                                                                                                                            (f_1348/client (+ (- 24) x_1350/client))
                                                                                                                            expansions:
                                                                                                                            • unroll
                                                                                                                              expr:
                                                                                                                              (f_1348/client (+ (- 25) x_1350/client))
                                                                                                                              expansions:
                                                                                                                              • unroll
                                                                                                                                expr:
                                                                                                                                (f_1348/client (+ (- 26) x_1350/client))
                                                                                                                                expansions:
                                                                                                                                • unroll
                                                                                                                                  expr:
                                                                                                                                  (f_1348/client (+ (- 27) x_1350/client))
                                                                                                                                  expansions:
                                                                                                                                  • unroll
                                                                                                                                    expr:
                                                                                                                                    (f_1348/client (+ (- 28) x_1350/client))
                                                                                                                                    expansions:
                                                                                                                                    • unroll
                                                                                                                                      expr:
                                                                                                                                      (f_1348/client (+ (- 29) x_1350/client))
                                                                                                                                      expansions:
                                                                                                                                      • unroll
                                                                                                                                        expr:
                                                                                                                                        (f_1348/client (+ (- 30) x_1350/client))
                                                                                                                                        expansions:
                                                                                                                                        • unroll
                                                                                                                                          expr:
                                                                                                                                          (f_1348/client (+ (- 31) x_1350/client))
                                                                                                                                          expansions:
                                                                                                                                          • unroll
                                                                                                                                            expr:
                                                                                                                                            (f_1348/client (+ (- 32) x_1350/client))
                                                                                                                                            expansions:
                                                                                                                                            • unroll
                                                                                                                                              expr:
                                                                                                                                              (f_1348/client (+ (- 33) x_1350/client))
                                                                                                                                              expansions:
                                                                                                                                              • unroll
                                                                                                                                                expr:
                                                                                                                                                (f_1348/client (+ (- 34) x_1350/client))
                                                                                                                                                expansions:
                                                                                                                                                • unroll
                                                                                                                                                  expr:
                                                                                                                                                  (f_1348/client (+ (- 35) x_1350/client))
                                                                                                                                                  expansions:
                                                                                                                                                  • unroll
                                                                                                                                                    expr:
                                                                                                                                                    (f_1348/client (+ (- 36) x_1350/client))
                                                                                                                                                    expansions:
                                                                                                                                                    • unroll
                                                                                                                                                      expr:
                                                                                                                                                      (f_1348/client (+ (- 37) x_1350/client))
                                                                                                                                                      expansions:
                                                                                                                                                      • unroll
                                                                                                                                                        expr:
                                                                                                                                                        (f_1348/client (+ (- 38) x_1350/client))
                                                                                                                                                        expansions:
                                                                                                                                                        • unroll
                                                                                                                                                          expr:
                                                                                                                                                          (f_1348/client (+ (- 39) x_1350/client))
                                                                                                                                                          expansions:
                                                                                                                                                          • unroll
                                                                                                                                                            expr:
                                                                                                                                                            (f_1348/client (+ (- 40) x_1350/client))
                                                                                                                                                            expansions:
                                                                                                                                                            • unroll
                                                                                                                                                              expr:
                                                                                                                                                              (f_1348/client (+ (- 41) x_1350/client))
                                                                                                                                                              expansions:
                                                                                                                                                              • unroll
                                                                                                                                                                expr:
                                                                                                                                                                (f_1348/client (+ (- 42) x_1350/client))
                                                                                                                                                                expansions:
                                                                                                                                                                • unroll
                                                                                                                                                                  expr:
                                                                                                                                                                  (f_1348/client (+ (- 43) x_1350/client))
                                                                                                                                                                  expansions:
                                                                                                                                                                  • unroll
                                                                                                                                                                    expr:
                                                                                                                                                                    (f_1348/client (+ (- 44) x_1350/client))
                                                                                                                                                                    expansions:
                                                                                                                                                                    • unroll
                                                                                                                                                                      expr:
                                                                                                                                                                      (f_1348/client (+ (- 45) x_1350/client))
                                                                                                                                                                      expansions:
                                                                                                                                                                      • unroll
                                                                                                                                                                        expr:
                                                                                                                                                                        (f_1348/client (+ (- 46) x_1350/client))
                                                                                                                                                                        expansions:
                                                                                                                                                                        • unroll
                                                                                                                                                                          expr:
                                                                                                                                                                          (f_1348/client (+ (- 47) x_1350/client))
                                                                                                                                                                          expansions:
                                                                                                                                                                          • unroll
                                                                                                                                                                            expr:
                                                                                                                                                                            (f_1348/client (+ (- 48) x_1350/client))
                                                                                                                                                                            expansions:
                                                                                                                                                                            • unroll
                                                                                                                                                                              expr:
                                                                                                                                                                              (f_1348/client (+ (- 49) x_1350/client))
                                                                                                                                                                              expansions:
                                                                                                                                                                              • unroll
                                                                                                                                                                                expr:
                                                                                                                                                                                (f_1348/client (+ (- 50) x_1350/client))
                                                                                                                                                                                expansions:
                                                                                                                                                                                • unroll
                                                                                                                                                                                  expr:
                                                                                                                                                                                  (f_1348/client (+ (- 51) x_1350/client))
                                                                                                                                                                                  expansions:
                                                                                                                                                                                  • unroll
                                                                                                                                                                                    expr:
                                                                                                                                                                                    (f_1348/client (+ (- 52) x_1350/client))
                                                                                                                                                                                    expansions:
                                                                                                                                                                                    • unroll
                                                                                                                                                                                      expr:
                                                                                                                                                                                      (f_1348/client (+ (- 53) x_1350/client))
                                                                                                                                                                                      expansions:
                                                                                                                                                                                      • unroll
                                                                                                                                                                                        expr:
                                                                                                                                                                                        (f_1348/client (+ (- 54) x_1350/client))
                                                                                                                                                                                        expansions:
                                                                                                                                                                                        • unroll
                                                                                                                                                                                          expr:
                                                                                                                                                                                          (f_1348/client (+ (- 55) x_1350/client))
                                                                                                                                                                                          expansions:
                                                                                                                                                                                          • unroll
                                                                                                                                                                                            expr:
                                                                                                                                                                                            (f_1348/client (+ (- 56) x_1350/client))
                                                                                                                                                                                            expansions:
                                                                                                                                                                                            • unroll
                                                                                                                                                                                              expr:
                                                                                                                                                                                              (f_1348/client (+ (- 57) x_1350/client))
                                                                                                                                                                                              expansions:
                                                                                                                                                                                              • unroll
                                                                                                                                                                                                expr:
                                                                                                                                                                                                (f_1348/client (+ (- 58) x_1350/client))
                                                                                                                                                                                                expansions:
                                                                                                                                                                                                • unroll
                                                                                                                                                                                                  expr:
                                                                                                                                                                                                  (f_1348/client (+ (- 59) x_1350/client))
                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                    expr:
                                                                                                                                                                                                    (f_1348/client (+ (- 60) x_1350/client))
                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                      expr:
                                                                                                                                                                                                      (f_1348/client (+ (- 61) x_1350/client))
                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                        expr:
                                                                                                                                                                                                        (f_1348/client (+ (- 62) x_1350/client))
                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                          expr:
                                                                                                                                                                                                          (f_1348/client (+ (- 63) x_1350/client))
                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                            expr:
                                                                                                                                                                                                            (f_1348/client (+ (- 64) x_1350/client))
                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                              expr:
                                                                                                                                                                                                              (f_1348/client (+ (- 65) x_1350/client))
                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                (f_1348/client (+ (- 66) x_1350/client))
                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                  (f_1348/client (+ (- 67) x_1350/client))
                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                    (f_1348/client (+ (- 68) x_1350/client))
                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                      (f_1348/client (+ (- 69) x_1350/client))
                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                        (f_1348/client (+ (- 70) x_1350/client))
                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                          (f_1348/client (+ (- 71) x_1350/client))
                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                            (f_1348/client (+ (- 72) x_1350/client))
                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                              (f_1348/client (+ (- 73) x_1350/client))
                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                (f_1348/client (+ (- 74) x_1350/client))
                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                  (f_1348/client (+ (- 75) x_1350/client))
                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                    (f_1348/client (+ (- 76) x_1350/client))
                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                      (f_1348/client (+ (- 77) x_1350/client))
                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                        (f_1348/client (+ (- 78) x_1350/client))
                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                          (f_1348/client (+ (- 79) x_1350/client))
                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                            (f_1348/client (+ (- 80) x_1350/client))
                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                              (f_1348/client (+ (- 81) x_1350/client))
                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                (f_1348/client (+ (- 82) x_1350/client))
                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                  (f_1348/client (+ (- 83) x_1350/client))
                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                    (f_1348/client (+ (- 84) x_1350/client))
                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                      (f_1348/client (+ (- 85) x_1350/client))
                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                        (f_1348/client (+ (- 86) x_1350/client))
                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                          (f_1348/client (+ (- 87) x_1350/client))
                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                            (f_1348/client (+ (- 88) x_1350/client))
                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                              (f_1348/client (+ (- 89) x_1350/client))
                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                (f_1348/client (+ (- 90) x_1350/client))
                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 91) x_1350/client))
                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 92) x_1350/client))
                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 93) x_1350/client))
                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 94) x_1350/client))
                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 95) x_1350/client))
                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 96) x_1350/client))
                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 97) x_1350/client))
                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 98) x_1350/client))
                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 99) x_1350/client))
                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                  • Unsat

                                                                                                                                                                                                                                                                                  But watch what happens if we ask Imandra to verify this for x < 101, thus exceding the number of recursive calls that Imandra unrolls by default:

                                                                                                                                                                                                                                                                                  In [5]:
                                                                                                                                                                                                                                                                                  verify (fun x -> x < 101 ==> f x = 1)
                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                  Out[5]:
                                                                                                                                                                                                                                                                                  - : Z.t -> bool = <fun>
                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                  Unknown (Verified up to bound 100)
                                                                                                                                                                                                                                                                                  Expand
                                                                                                                                                                                                                                                                                  expanded:
                                                                                                                                                                                                                                                                                  • f ((-28) + x)
                                                                                                                                                                                                                                                                                  • f ((-70) + x)
                                                                                                                                                                                                                                                                                  • f ((-49) + x)
                                                                                                                                                                                                                                                                                  • f ((-72) + x)
                                                                                                                                                                                                                                                                                  • f ((-74) + x)
                                                                                                                                                                                                                                                                                  • f ((-36) + x)
                                                                                                                                                                                                                                                                                  • f ((-87) + x)
                                                                                                                                                                                                                                                                                  • f ((-56) + x)
                                                                                                                                                                                                                                                                                  • f ((-55) + x)
                                                                                                                                                                                                                                                                                  • f ((-33) + x)
                                                                                                                                                                                                                                                                                  • f ((-11) + x)
                                                                                                                                                                                                                                                                                  • f ((-95) + x)
                                                                                                                                                                                                                                                                                  • f ((-20) + x)
                                                                                                                                                                                                                                                                                  • f ((-12) + x)
                                                                                                                                                                                                                                                                                  • f ((-88) + x)
                                                                                                                                                                                                                                                                                  • f ((-65) + x)
                                                                                                                                                                                                                                                                                  • f ((-39) + x)
                                                                                                                                                                                                                                                                                  • f ((-25) + x)
                                                                                                                                                                                                                                                                                  • f ((-91) + x)
                                                                                                                                                                                                                                                                                  • f ((-35) + x)
                                                                                                                                                                                                                                                                                  • f ((-7) + x)
                                                                                                                                                                                                                                                                                  • f ((-90) + x)
                                                                                                                                                                                                                                                                                  • f ((-77) + x)
                                                                                                                                                                                                                                                                                  • f ((-57) + x)
                                                                                                                                                                                                                                                                                  • f ((-26) + x)
                                                                                                                                                                                                                                                                                  • f ((-86) + x)
                                                                                                                                                                                                                                                                                  • f ((-62) + x)
                                                                                                                                                                                                                                                                                  • f ((-42) + x)
                                                                                                                                                                                                                                                                                  • f ((-9) + x)
                                                                                                                                                                                                                                                                                  • f ((-16) + x)
                                                                                                                                                                                                                                                                                  • f ((-76) + x)
                                                                                                                                                                                                                                                                                  • f ((-61) + x)
                                                                                                                                                                                                                                                                                  • f ((-18) + x)
                                                                                                                                                                                                                                                                                  • f ((-46) + x)
                                                                                                                                                                                                                                                                                  • f ((-37) + x)
                                                                                                                                                                                                                                                                                  • f ((-31) + x)
                                                                                                                                                                                                                                                                                  • f ((-99) + x)
                                                                                                                                                                                                                                                                                  • f ((-34) + x)
                                                                                                                                                                                                                                                                                  • f ((-27) + x)
                                                                                                                                                                                                                                                                                  • f ((-80) + x)
                                                                                                                                                                                                                                                                                  • f ((-66) + x)
                                                                                                                                                                                                                                                                                  • f ((-89) + x)
                                                                                                                                                                                                                                                                                  • f ((-3) + x)
                                                                                                                                                                                                                                                                                  • f ((-63) + x)
                                                                                                                                                                                                                                                                                  • f ((-41) + x)
                                                                                                                                                                                                                                                                                  • f ((-13) + x)
                                                                                                                                                                                                                                                                                  • f ((-54) + x)
                                                                                                                                                                                                                                                                                  • f x
                                                                                                                                                                                                                                                                                  • f ((-92) + x)
                                                                                                                                                                                                                                                                                  • f ((-5) + x)
                                                                                                                                                                                                                                                                                  • f ((-79) + x)
                                                                                                                                                                                                                                                                                  • f ((-1) + x)
                                                                                                                                                                                                                                                                                  • f ((-97) + x)
                                                                                                                                                                                                                                                                                  • f ((-84) + x)
                                                                                                                                                                                                                                                                                  • f ((-93) + x)
                                                                                                                                                                                                                                                                                  • f ((-71) + x)
                                                                                                                                                                                                                                                                                  • f ((-94) + x)
                                                                                                                                                                                                                                                                                  • f ((-85) + x)
                                                                                                                                                                                                                                                                                  • f ((-60) + x)
                                                                                                                                                                                                                                                                                  • f ((-43) + x)
                                                                                                                                                                                                                                                                                  • f ((-78) + x)
                                                                                                                                                                                                                                                                                  • f ((-47) + x)
                                                                                                                                                                                                                                                                                  • f ((-51) + x)
                                                                                                                                                                                                                                                                                  • f ((-53) + x)
                                                                                                                                                                                                                                                                                  • f ((-8) + x)
                                                                                                                                                                                                                                                                                  • f ((-59) + x)
                                                                                                                                                                                                                                                                                  • f ((-30) + x)
                                                                                                                                                                                                                                                                                  • f ((-6) + x)
                                                                                                                                                                                                                                                                                  • f ((-15) + x)
                                                                                                                                                                                                                                                                                  • f ((-40) + x)
                                                                                                                                                                                                                                                                                  • f ((-29) + x)
                                                                                                                                                                                                                                                                                  • f ((-22) + x)
                                                                                                                                                                                                                                                                                  • f ((-10) + x)
                                                                                                                                                                                                                                                                                  • f ((-4) + x)
                                                                                                                                                                                                                                                                                  • f ((-75) + x)
                                                                                                                                                                                                                                                                                  • f ((-38) + x)
                                                                                                                                                                                                                                                                                  • f ((-24) + x)
                                                                                                                                                                                                                                                                                  • f ((-45) + x)
                                                                                                                                                                                                                                                                                  • f ((-19) + x)
                                                                                                                                                                                                                                                                                  • f ((-73) + x)
                                                                                                                                                                                                                                                                                  • f ((-83) + x)
                                                                                                                                                                                                                                                                                  • f ((-82) + x)
                                                                                                                                                                                                                                                                                  • f ((-48) + x)
                                                                                                                                                                                                                                                                                  • f ((-68) + x)
                                                                                                                                                                                                                                                                                  • f ((-67) + x)
                                                                                                                                                                                                                                                                                  • f ((-21) + x)
                                                                                                                                                                                                                                                                                  • f ((-98) + x)
                                                                                                                                                                                                                                                                                  • f ((-50) + x)
                                                                                                                                                                                                                                                                                  • f ((-14) + x)
                                                                                                                                                                                                                                                                                  • f ((-69) + x)
                                                                                                                                                                                                                                                                                  • f ((-81) + x)
                                                                                                                                                                                                                                                                                  • f ((-58) + x)
                                                                                                                                                                                                                                                                                  • f ((-17) + x)
                                                                                                                                                                                                                                                                                  • f ((-64) + x)
                                                                                                                                                                                                                                                                                  • f ((-44) + x)
                                                                                                                                                                                                                                                                                  • f ((-96) + x)
                                                                                                                                                                                                                                                                                  • f ((-32) + x)
                                                                                                                                                                                                                                                                                  • f ((-23) + x)
                                                                                                                                                                                                                                                                                  • f ((-2) + x)
                                                                                                                                                                                                                                                                                  • f ((-52) + x)
                                                                                                                                                                                                                                                                                  blocked:
                                                                                                                                                                                                                                                                                  • f ((-100) + x)
                                                                                                                                                                                                                                                                                  proof attempt
                                                                                                                                                                                                                                                                                  ground_instances:100
                                                                                                                                                                                                                                                                                  definitions:0
                                                                                                                                                                                                                                                                                  inductions:0
                                                                                                                                                                                                                                                                                  search_time:
                                                                                                                                                                                                                                                                                  0.161s
                                                                                                                                                                                                                                                                                  Expand
                                                                                                                                                                                                                                                                                  • start[0.161s] ( :var_0: ) < 101 ==> f ( :var_0: ) = 1
                                                                                                                                                                                                                                                                                  • simplify

                                                                                                                                                                                                                                                                                    into:
                                                                                                                                                                                                                                                                                    101 <= ( :var_0: ) || f ( :var_0: ) = 1
                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                    []
                                                                                                                                                                                                                                                                                    rewrite_steps:
                                                                                                                                                                                                                                                                                      forward_chaining:
                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                        (f_1348/client x_1352/client)
                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 1) x_1352/client))
                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 2) x_1352/client))
                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 3) x_1352/client))
                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 4) x_1352/client))
                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 5) x_1352/client))
                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 6) x_1352/client))
                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 7) x_1352/client))
                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 8) x_1352/client))
                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 9) x_1352/client))
                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 10) x_1352/client))
                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 11) x_1352/client))
                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 12) x_1352/client))
                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 13) x_1352/client))
                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 14) x_1352/client))
                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 15) x_1352/client))
                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 16) x_1352/client))
                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 17) x_1352/client))
                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 18) x_1352/client))
                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 19) x_1352/client))
                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 20) x_1352/client))
                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 21) x_1352/client))
                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 22) x_1352/client))
                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 23) x_1352/client))
                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 24) x_1352/client))
                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 25) x_1352/client))
                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 26) x_1352/client))
                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 27) x_1352/client))
                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 28) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 29) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 30) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 31) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 32) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 33) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 34) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 35) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 36) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 37) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 38) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 39) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 40) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 41) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 42) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 43) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 44) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 45) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 46) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 47) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 48) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 49) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 50) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 51) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 52) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 53) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 54) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 55) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 56) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 57) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 58) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 59) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 60) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 61) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 62) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 63) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 64) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 65) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 66) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 67) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 68) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 69) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 70) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 71) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 72) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 73) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 74) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 75) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 76) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 77) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 78) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 79) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 80) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 81) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 82) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 83) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 84) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 85) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 86) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 87) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 88) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 89) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 90) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 91) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (f_1348/client (+ (- 92) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (f_1348/client (+ (- 93) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (f_1348/client (+ (- 94) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (f_1348/client (+ (- 95) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (f_1348/client (+ (- 96) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (f_1348/client (+ (- 97) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (f_1348/client (+ (- 98) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            expansions:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            • unroll
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expr:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (f_1348/client (+ (- 99) x_1352/client))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              expansions:

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              As expected, since the recursion depth needed to prove this exceeds the unrolling bound we set, Imandra could only prove this property up to bound k. This goal is in fact a property that is better suited for verification by induction (indeed, you might try adding the [@@auto] annotation to the above goal to invoke the Imandra's inductive waterfall and prove it).

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              As a minor note, if we reach a local unrolling depth instead of hitting the global one, Imandra will be a bit more positive in its message, telling us that the conjecture has been proved up to the number of steps we've specified instead of a weaker "Unknown":

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              In [6]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              verify ~upto:100 (fun x -> x < 101 ==> f x = 1)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Out[6]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              - : Z.t -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Proved up to 100 steps

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Datatype bounds

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Imandra offers an orthogonal, more advanced form of bounding for unrolling, using ~upto_bound:<n>. This bound works in a fundamentally different way than the "normal" unrolling limit: instead of acting as a limit on the number of recursive steps that Imandra unrolls, this limit instructs Imandra to synthesize a recursive depth function for the datatypes involved in the goal and to transform the goal such that it will include this bound check as part of the hypotheses.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              This means that Imandra will be actually proving a bounded theorem involving the depth of datatypes.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              In [7]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              verify ~upto_bound:5 (fun x -> List.rev (List.rev x) = x)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Out[7]:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              - : 'a list -> bool = <fun>
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Proved up to bound 5

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              When using unrolling with ~upto_bound, one should remember that the global unrolling limit still applies and may cause Imandra to abort solving before reaching the datatype bound.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              This datatype bound limit can also be used with the blast strategy, while the unrolling limit can't, as blast doesn't work by recursive unrolling.