/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 623 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 653 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalfstart(A, B, C) -> Com_1(evalfentryin(A, B, C)) :|: TRUE evalfentryin(A, B, C) -> Com_1(evalfbb3in(C, B, A)) :|: A >= 1 && B >= A + 1 evalfbb3in(A, B, C) -> Com_1(evalfbbin(A, B, C)) :|: C >= 1 && B >= C + 1 evalfbb3in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: 0 >= C evalfbb3in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: C >= B evalfbbin(A, B, C) -> Com_1(evalfbb1in(A, B, C)) :|: A >= 1 evalfbbin(A, B, C) -> Com_1(evalfbb2in(A, B, C)) :|: 0 >= A evalfbb1in(A, B, C) -> Com_1(evalfbb3in(A, B, C + 1)) :|: TRUE evalfbb2in(A, B, C) -> Com_1(evalfbb3in(A, B, C - 1)) :|: TRUE evalfreturnin(A, B, C) -> Com_1(evalfstop(A, B, C)) :|: TRUE The start-symbols are:[evalfstart_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 3*ar_0 + 6*ar_1 + 15) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) (Comp: ?, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) (Comp: ?, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfstart) = 2 Pol(evalfentryin) = 2 Pol(evalfbb3in) = 2 Pol(evalfbbin) = 2 Pol(evalfreturnin) = 1 Pol(evalfbb1in) = 2 Pol(evalfbb2in) = 2 Pol(evalfstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ] evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol evalfbb1in: X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_1 + X_3 - 2 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ X_1 - 1 >= 0 For symbol evalfbb2in: X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ -X_1 + X_2 - 2 >= 0 /\ -X_1 >= 0 For symbol evalfbb3in: X_2 - 2 >= 0 For symbol evalfbbin: X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_2 - 2 >= 0 For symbol evalfreturnin: X_2 - 2 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 By chaining the transition koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] with all transitions in problem 4, the following new transition is obtained: koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] We thus obtain the following problem: 5: T: (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 5: evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) We thus obtain the following problem: 6: T: (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] with all transitions in problem 6, the following new transition is obtained: evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] We thus obtain the following problem: 7: T: (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] with all transitions in problem 7, the following new transition is obtained: evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] We thus obtain the following problem: 8: T: (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 8: evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 ] We thus obtain the following problem: 9: T: (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 ] with all transitions in problem 9, the following new transition is obtained: evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] We thus obtain the following problem: 10: T: (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 10: evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] We thus obtain the following problem: 11: T: (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 ] with all transitions in problem 11, the following new transition is obtained: evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] We thus obtain the following problem: 12: T: (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 12: evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] We thus obtain the following problem: 13: T: (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] with all transitions in problem 13, the following new transition is obtained: evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] We thus obtain the following problem: 14: T: (Comp: 1, Cost: 2) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2)) [ 0 <= 0 ] with all transitions in problem 14, the following new transition is obtained: koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] We thus obtain the following problem: 15: T: (Comp: 1, Cost: 3) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] (Comp: 1, Cost: 2) evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 15: evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] We thus obtain the following problem: 16: T: (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 1, Cost: 3) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 ] with all transitions in problem 16, the following new transitions are obtained: evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] We thus obtain the following problem: 17: T: (Comp: ?, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 1, Cost: 3) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 17 produces the following problem: 18: T: (Comp: ?, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: ?, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 1, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: 1, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 1, Cost: 3) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfbb3in) = V_2 - V_3 and size complexities S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\\ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 /\\ ar_1 - 2 >= 0 ]", 0-0) = ar_2 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\\ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 /\\ ar_1 - 2 >= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\\ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 /\\ ar_1 - 2 >= 0 ]", 0-2) = ar_0 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_2 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-0) = ar_2 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-1) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-2) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ ar_2 >= ar_1 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ ar_2 >= ar_1 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ ar_2 >= ar_1 ]", 0-2) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ 0 >= ar_2 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ 0 >= ar_2 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ 0 >= ar_2 ]", 0-2) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-2) = ar_0 orients the transitions evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] weakly and the transition evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] strictly and produces the following problem: 19: T: (Comp: ?, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: 2*ar_1, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 1, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: 1, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 1, Cost: 3) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfbb3in) = V_3 and size complexities S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\\ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 /\\ ar_1 - 2 >= 0 ]", 0-0) = ar_2 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\\ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 /\\ ar_1 - 2 >= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\\ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 /\\ ar_1 - 2 >= 0 ]", 0-2) = ar_0 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_2 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-0) = ar_2 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-1) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_1 - 2 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-2) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ ar_2 >= ar_1 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ ar_2 >= ar_1 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ ar_2 >= ar_1 ]", 0-2) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ 0 >= ar_2 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ 0 >= ar_2 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\\ 0 >= ar_2 ]", 0-2) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 >= 1 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-0) = ar_2 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\\ ar_2 >= 1 /\\ ar_1 >= ar_2 + 1 /\\ ar_1 - ar_2 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ 0 >= ar_0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_1 - 2 >= 0 /\\ -ar_0 >= 0 ]", 0-2) = ar_0 orients the transitions evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] weakly and the transition evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] strictly and produces the following problem: 20: T: (Comp: ar_0, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: 2*ar_1, Cost: 3) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - 2 >= 0 /\ ar_2 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ 0 >= ar_2 ] (Comp: 2, Cost: 2) evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 2 >= 0 /\ ar_2 >= ar_1 ] (Comp: 1, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 - 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ 0 >= ar_0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_1 - 2 >= 0 /\ -ar_0 >= 0 ] (Comp: 1, Cost: 2) evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 >= 1 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 1, Cost: 3) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_2, ar_1, ar_0)) [ 0 <= 0 /\ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 /\ ar_1 - 2 >= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 3*ar_0 + 6*ar_1 + 15 Time: 0.650 sec (SMT: 0.534 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb3in : A'=C, C'=A, [ A>=1 && B>=1+A ], cost: 1 2: evalfbb3in -> evalfbbin : [ C>=1 && B>=1+C ], cost: 1 3: evalfbb3in -> evalfreturnin : [ 0>=C ], cost: 1 4: evalfbb3in -> evalfreturnin : [ C>=B ], cost: 1 5: evalfbbin -> evalfbb1in : [ A>=1 ], cost: 1 6: evalfbbin -> evalfbb2in : [ 0>=A ], cost: 1 7: evalfbb1in -> evalfbb3in : C'=1+C, [], cost: 1 8: evalfbb2in -> evalfbb3in : C'=-1+C, [], cost: 1 9: evalfreturnin -> evalfstop : [], cost: 1 Removed unreachable and leaf rules: Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb3in : A'=C, C'=A, [ A>=1 && B>=1+A ], cost: 1 2: evalfbb3in -> evalfbbin : [ C>=1 && B>=1+C ], cost: 1 5: evalfbbin -> evalfbb1in : [ A>=1 ], cost: 1 6: evalfbbin -> evalfbb2in : [ 0>=A ], cost: 1 7: evalfbb1in -> evalfbb3in : C'=1+C, [], cost: 1 8: evalfbb2in -> evalfbb3in : C'=-1+C, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalfstart 10: evalfstart -> evalfbb3in : A'=C, C'=A, [ A>=1 && B>=1+A ], cost: 2 2: evalfbb3in -> evalfbbin : [ C>=1 && B>=1+C ], cost: 1 11: evalfbbin -> evalfbb3in : C'=1+C, [ A>=1 ], cost: 2 12: evalfbbin -> evalfbb3in : C'=-1+C, [ 0>=A ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: evalfstart 10: evalfstart -> evalfbb3in : A'=C, C'=A, [ A>=1 && B>=1+A ], cost: 2 13: evalfbb3in -> evalfbb3in : C'=1+C, [ C>=1 && B>=1+C && A>=1 ], cost: 3 14: evalfbb3in -> evalfbb3in : C'=-1+C, [ C>=1 && B>=1+C && 0>=A ], cost: 3 Accelerating simple loops of location 2. Accelerating the following rules: 13: evalfbb3in -> evalfbb3in : C'=1+C, [ C>=1 && B>=1+C && A>=1 ], cost: 3 14: evalfbb3in -> evalfbb3in : C'=-1+C, [ C>=1 && B>=1+C && 0>=A ], cost: 3 Accelerated rule 13 with metering function -C+B, yielding the new rule 15. Accelerated rule 14 with metering function C, yielding the new rule 16. Removing the simple loops: 13 14. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 10: evalfstart -> evalfbb3in : A'=C, C'=A, [ A>=1 && B>=1+A ], cost: 2 15: evalfbb3in -> evalfbb3in : C'=B, [ C>=1 && B>=1+C && A>=1 ], cost: -3*C+3*B 16: evalfbb3in -> evalfbb3in : C'=0, [ C>=1 && B>=1+C && 0>=A ], cost: 3*C Chained accelerated rules (with incoming rules): Start location: evalfstart 10: evalfstart -> evalfbb3in : A'=C, C'=A, [ A>=1 && B>=1+A ], cost: 2 17: evalfstart -> evalfbb3in : A'=C, C'=B, [ A>=1 && B>=1+A && C>=1 ], cost: 2-3*A+3*B 18: evalfstart -> evalfbb3in : A'=C, C'=0, [ A>=1 && B>=1+A && 0>=C ], cost: 2+3*A Removed unreachable locations (and leaf rules with constant cost): Start location: evalfstart 17: evalfstart -> evalfbb3in : A'=C, C'=B, [ A>=1 && B>=1+A && C>=1 ], cost: 2-3*A+3*B 18: evalfstart -> evalfbb3in : A'=C, C'=0, [ A>=1 && B>=1+A && 0>=C ], cost: 2+3*A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalfstart 17: evalfstart -> evalfbb3in : A'=C, C'=B, [ A>=1 && B>=1+A && C>=1 ], cost: 2-3*A+3*B 18: evalfstart -> evalfbb3in : A'=C, C'=0, [ A>=1 && B>=1+A && 0>=C ], cost: 2+3*A Computing asymptotic complexity for rule 17 Solved the limit problem by the following transformations: Created initial limit problem: 2-3*A+3*B (+), C (+/+!), A (+/+!), -A+B (+/+!) [not solved] applying transformation rule (C) using substitution {A==1} resulting limit problem: 1 (+/+!), C (+/+!), -1+B (+/+!), -1+3*B (+) [not solved] applying transformation rule (C) using substitution {B==1+A} resulting limit problem: 1 (+/+!), C (+/+!), A (+/+!), 2+3*A (+) [not solved] applying transformation rule (C) using substitution {C==1} resulting limit problem: 1 (+/+!), A (+/+!), 2+3*A (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: A (+/+!), 2+3*A (+) [not solved] applying transformation rule (D), replacing 2+3*A (+) by 3*A (+) resulting limit problem: 3*A (+), A (+/+!) [not solved] applying transformation rule (A), replacing 3*A (+) by A (+) and 3 (+!) using + limit vector (+,+!) resulting limit problem: 3 (+!), A (+) [not solved] applying transformation rule (B), deleting 3 (+!) resulting limit problem: A (+) [solved] Solution: C / 1 A / 1 B / 1+n Resulting cost 2+3*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 2+3*n Rule cost: 2-3*A+3*B Rule guard: [ A>=1 && B>=1+A && C>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)