/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 424 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 822 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalfstart(A, B, C, D) -> Com_1(evalfentryin(A, B, C, D)) :|: TRUE evalfentryin(A, B, C, D) -> Com_1(evalfbb3in(A, B, 0, 0)) :|: A >= 1 && B >= A + 1 evalfbb3in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: D >= B evalfbb3in(A, B, C, D) -> Com_1(evalfbb4in(A, B, C, D)) :|: B >= D + 1 evalfbb4in(A, B, C, D) -> Com_1(evalfbbin(A, B, C, D)) :|: 0 >= E + 1 evalfbb4in(A, B, C, D) -> Com_1(evalfbbin(A, B, C, D)) :|: E >= 1 evalfbb4in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: TRUE evalfbbin(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: A >= C + 1 evalfbbin(A, B, C, D) -> Com_1(evalfbb2in(A, B, C, D)) :|: C >= A evalfbb1in(A, B, C, D) -> Com_1(evalfbb3in(A, B, C + 1, D)) :|: TRUE evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(A, B, 0, D + 1)) :|: TRUE evalfreturnin(A, B, C, D) -> Com_1(evalfstop(A, B, C, D)) :|: TRUE The start-symbols are:[evalfstart_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 20*ar_1 + 10*ar_0 + 20*ar_0*ar_1 + 16) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) (Comp: ?, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 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, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) (Comp: ?, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfstart) = 2 Pol(evalfentryin) = 2 Pol(evalfbb3in) = 2 Pol(evalfreturnin) = 1 Pol(evalfbb4in) = 2 Pol(evalfbbin) = 2 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, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] (Comp: 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 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_4 - 1 >= 0 /\ X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 - 2 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalfbb2in: X_2 - X_4 - 1 >= 0 /\ X_4 >= 0 /\ X_3 + X_4 - 1 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_1 + X_3 - 2 >= 0 /\ -X_1 + X_3 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalfbb3in: X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalfbb4in: X_2 - X_4 - 1 >= 0 /\ X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalfbbin: X_2 - X_4 - 1 >= 0 /\ X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalfreturnin: X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 - 1 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ -ar_0 + ar_2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_1 - ar_2 - 2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_2 >= ar_0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_2 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ e >= 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= e + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_3 + 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_3 >= ar_1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 2*V_2 Pol(evalfstart) = 2*V_2 Pol(evalfreturnin) = 2*V_2 - 2*V_4 Pol(evalfstop) = 2*V_2 - 2*V_4 Pol(evalfbb2in) = 2*V_2 - 2*V_4 - 1 Pol(evalfbb3in) = 2*V_2 - 2*V_4 Pol(evalfbb1in) = 2*V_2 - 2*V_4 Pol(evalfbbin) = 2*V_2 - 2*V_4 Pol(evalfbb4in) = 2*V_2 - 2*V_4 Pol(evalfentryin) = 2*V_2 orients all transitions weakly and the transitions evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_2 >= ar_0 ] evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 - 1 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ -ar_0 + ar_2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_1, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 - 1 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ -ar_0 + ar_2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_1 - ar_2 - 2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_1, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_2 >= ar_0 ] (Comp: ?, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_2 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ e >= 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= e + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_3 + 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_3 >= ar_1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfbbin) = 2*V_1 - 2*V_3 + 1 Pol(evalfbb1in) = 2*V_1 - 2*V_3 Pol(evalfbb4in) = 2*V_1 - 2*V_3 + 1 Pol(evalfbb3in) = 2*V_1 - 2*V_3 + 1 and size complexities S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 ]", 0-0) = ar_0 S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 ]", 0-2) = 0 S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\\ ar_1 >= ar_0 + 1 ]", 0-3) = 0 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_3 >= ar_1 ]", 0-0) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_3 >= ar_1 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_3 >= ar_1 ]", 0-2) = 0 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_3 >= ar_1 ]", 0-3) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_3 + 1 ]", 0-0) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_3 + 1 ]", 0-1) = ar_1 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_3 + 1 ]", 0-2) = ar_0 S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_3 + 1 ]", 0-3) = ar_1 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= e + 1 ]", 0-0) = ar_0 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= e + 1 ]", 0-1) = ar_1 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= e + 1 ]", 0-2) = ar_0 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= e + 1 ]", 0-3) = ar_1 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ e >= 1 ]", 0-0) = ar_0 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ e >= 1 ]", 0-1) = ar_1 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ e >= 1 ]", 0-2) = ar_0 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ e >= 1 ]", 0-3) = ar_1 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_0 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_0 S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-3) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_2 + 1 ]", 0-0) = ar_0 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_2 + 1 ]", 0-1) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_2 + 1 ]", 0-2) = ar_0 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_2 + 1 ]", 0-3) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_2 >= ar_0 ]", 0-0) = ar_0 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_2 >= ar_0 ]", 0-1) = ar_1 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_2 >= ar_0 ]", 0-2) = ar_0 S("evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_2 >= ar_0 ]", 0-3) = ar_1 S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_1 - ar_2 - 2 >= 0 /\\ ar_0 - ar_2 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_0 S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_1 - ar_2 - 2 >= 0 /\\ ar_0 - ar_2 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_1 - ar_2 - 2 >= 0 /\\ ar_0 - ar_2 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_0 S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_1 - ar_2 - 2 >= 0 /\\ ar_0 - ar_2 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-3) = ar_1 S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 - 1 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ -ar_0 + ar_2 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_0 S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 - 1 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ -ar_0 + ar_2 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 - 1 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ -ar_0 + ar_2 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = 0 S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\\ ar_3 >= 0 /\\ ar_2 + ar_3 - 1 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 3 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ -ar_0 + ar_2 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-3) = ar_1 S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_0 S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_1 S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_0 S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\\ ar_2 + ar_3 >= 0 /\\ ar_1 + ar_3 - 2 >= 0 /\\ ar_0 + ar_3 - 1 >= 0 /\\ ar_2 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 - 2 >= 0 /\\ ar_0 + ar_1 - 3 >= 0 /\\ -ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-3) = ar_1 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 orients the transitions evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_2 + 1 ] evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ e >= 1 ] evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= e + 1 ] evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_3 + 1 ] evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_1 - ar_2 - 2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] weakly and the transitions evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_2 + 1 ] evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_1 - ar_2 - 2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_1, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 - 1 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ -ar_0 + ar_2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 2*ar_1 + 1, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_1 - ar_2 - 2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_1, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_2 >= ar_0 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 2*ar_1 + 1, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_2 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ e >= 1 ] (Comp: ?, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= e + 1 ] (Comp: ?, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_3 + 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_3 >= ar_1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 6 produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_1, Cost: 1) evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, ar_3 + 1)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 - 1 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 3 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ -ar_0 + ar_2 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 2*ar_1 + 1, Cost: 1) evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2 + 1, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_1 - ar_2 - 2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_1, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_2 >= ar_0 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 2*ar_1 + 1, Cost: 1) evalfbbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_2 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 4*ar_1 + 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ e >= 1 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 4*ar_1 + 2, Cost: 1) evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbbin(ar_0, ar_1, ar_2, ar_3)) [ ar_1 - ar_3 - 1 >= 0 /\ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= e + 1 ] (Comp: 2*ar_0 + 4*ar_0*ar_1 + 4*ar_1 + 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_3 + 1 ] (Comp: 2, Cost: 1) evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= 0 /\ ar_2 + ar_3 >= 0 /\ ar_1 + ar_3 - 2 >= 0 /\ ar_0 + ar_3 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 - 2 >= 0 /\ ar_0 + ar_1 - 3 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_3 >= ar_1 ] (Comp: 1, Cost: 1) evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, 0, 0)) [ ar_0 >= 1 /\ ar_1 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3)) start location: koat_start leaf cost: 0 Complexity upper bound 20*ar_1 + 10*ar_0 + 20*ar_0*ar_1 + 16 Time: 0.411 sec (SMT: 0.326 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 1 2: evalfbb3in -> evalfreturnin : [ D>=B ], cost: 1 3: evalfbb3in -> evalfbb4in : [ B>=1+D ], cost: 1 4: evalfbb4in -> evalfbbin : [ 0>=1+free ], cost: 1 5: evalfbb4in -> evalfbbin : [ free_1>=1 ], cost: 1 6: evalfbb4in -> evalfreturnin : [], cost: 1 7: evalfbbin -> evalfbb1in : [ A>=1+C ], cost: 1 8: evalfbbin -> evalfbb2in : [ C>=A ], cost: 1 9: evalfbb1in -> evalfbb3in : C'=1+C, [], cost: 1 10: evalfbb2in -> evalfbb3in : C'=0, D'=1+D, [], cost: 1 11: evalfreturnin -> evalfstop : [], cost: 1 Removed unreachable and leaf rules: Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 1 3: evalfbb3in -> evalfbb4in : [ B>=1+D ], cost: 1 4: evalfbb4in -> evalfbbin : [ 0>=1+free ], cost: 1 5: evalfbb4in -> evalfbbin : [ free_1>=1 ], cost: 1 7: evalfbbin -> evalfbb1in : [ A>=1+C ], cost: 1 8: evalfbbin -> evalfbb2in : [ C>=A ], cost: 1 9: evalfbb1in -> evalfbb3in : C'=1+C, [], cost: 1 10: evalfbb2in -> evalfbb3in : C'=0, D'=1+D, [], cost: 1 Simplified all rules, resulting in: Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 1 3: evalfbb3in -> evalfbb4in : [ B>=1+D ], cost: 1 5: evalfbb4in -> evalfbbin : [], cost: 1 7: evalfbbin -> evalfbb1in : [ A>=1+C ], cost: 1 8: evalfbbin -> evalfbb2in : [ C>=A ], cost: 1 9: evalfbb1in -> evalfbb3in : C'=1+C, [], cost: 1 10: evalfbb2in -> evalfbb3in : C'=0, D'=1+D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalfstart 12: evalfstart -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 2 13: evalfbb3in -> evalfbbin : [ B>=1+D ], cost: 2 14: evalfbbin -> evalfbb3in : C'=1+C, [ A>=1+C ], cost: 2 15: evalfbbin -> evalfbb3in : C'=0, D'=1+D, [ C>=A ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: evalfstart 12: evalfstart -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 2 16: evalfbb3in -> evalfbb3in : C'=1+C, [ B>=1+D && A>=1+C ], cost: 4 17: evalfbb3in -> evalfbb3in : C'=0, D'=1+D, [ B>=1+D && C>=A ], cost: 4 Accelerating simple loops of location 2. Accelerating the following rules: 16: evalfbb3in -> evalfbb3in : C'=1+C, [ B>=1+D && A>=1+C ], cost: 4 17: evalfbb3in -> evalfbb3in : C'=0, D'=1+D, [ B>=1+D && C>=A ], cost: 4 Accelerated rule 16 with metering function -C+A, yielding the new rule 18. Accelerated rule 17 with metering function -D+B (after strengthening guard), yielding the new rule 19. Nested simple loops 17 (outer loop) and 18 (inner loop) with metering function -1-D+B, resulting in the new rules: 20, 21. Removing the simple loops: 16 17. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 12: evalfstart -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 2 18: evalfbb3in -> evalfbb3in : C'=A, [ B>=1+D && A>=1+C ], cost: -4*C+4*A 19: evalfbb3in -> evalfbb3in : C'=0, D'=B, [ B>=1+D && C>=A && 0>=A ], cost: -4*D+4*B 20: evalfbb3in -> evalfbb3in : C'=A, D'=-1+B, [ C>=A && B>=2+D && A>=1 ], cost: -4-4*(1+D-B)*A-4*D+4*B 21: evalfbb3in -> evalfbb3in : C'=A, D'=-1+B, [ A>=1+C && B>=2+D && A>=1 ], cost: -4-4*C-4*(1+D-B)*A-4*D+4*A+4*B Chained accelerated rules (with incoming rules): Start location: evalfstart 12: evalfstart -> evalfbb3in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 2 22: evalfstart -> evalfbb3in : C'=A, D'=0, [ A>=1 && B>=1+A && B>=1 ], cost: 2+4*A 23: evalfstart -> evalfbb3in : C'=A, D'=-1+B, [ A>=1 && B>=1+A && B>=2 ], cost: -2+4*A+4*(-1+B)*A+4*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalfstart 22: evalfstart -> evalfbb3in : C'=A, D'=0, [ A>=1 && B>=1+A && B>=1 ], cost: 2+4*A 23: evalfstart -> evalfbb3in : C'=A, D'=-1+B, [ A>=1 && B>=1+A && B>=2 ], cost: -2+4*A+4*(-1+B)*A+4*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalfstart 22: evalfstart -> evalfbb3in : C'=A, D'=0, [ A>=1 && B>=1+A && B>=1 ], cost: 2+4*A 23: evalfstart -> evalfbb3in : C'=A, D'=-1+B, [ A>=1 && B>=1+A && B>=2 ], cost: -2+4*A+4*(-1+B)*A+4*B Computing asymptotic complexity for rule 22 Solved the limit problem by the following transformations: Created initial limit problem: 2+4*A (+), A (+/+!), -A+B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n,B==2*n} resulting limit problem: [solved] Solution: A / n B / 2*n Resulting cost 2+4*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 23 Solved the limit problem by the following transformations: Created initial limit problem: -2+4*A*B+4*B (+), A (+/+!), -A+B (+/+!) [not solved] applying transformation rule (C) using substitution {A==1} resulting limit problem: 1 (+/+!), -2+8*B (+), -1+B (+/+!) [not solved] applying transformation rule (C) using substitution {B==1+A} resulting limit problem: 1 (+/+!), 6+8*A (+), A (+/+!) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 6+8*A (+), A (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n} resulting limit problem: [solved] Solved the limit problem by the following transformations: Created initial limit problem: -2+4*A*B+4*B (+), A (+/+!), -A+B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==1+n,B==2+n} resulting limit problem: [solved] Solution: A / 1+n B / 2+n Resulting cost 14+16*n+4*n^2 has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 14+16*n+4*n^2 Rule cost: -2+4*A+4*(-1+B)*A+4*B Rule guard: [ A>=1 && B>=1+A ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)