/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 341 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 1442 ms] (4) BOUNDS(n^1, 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(evalfbb5in(B, B, C, D)) :|: TRUE evalfbb5in(A, B, C, D) -> Com_1(evalfbbin(A, B, C, D)) :|: B >= 2 evalfbb5in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: 1 >= B evalfbbin(A, B, C, D) -> Com_1(evalfbb2in(A, B, B - 1, A + B - 1)) :|: TRUE evalfbb2in(A, B, C, D) -> Com_1(evalfbb4in(A, B, C, D)) :|: C >= D + 1 evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(A, B, C, D)) :|: D >= C evalfbb3in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: 0 >= E + 1 evalfbb3in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: E >= 1 evalfbb3in(A, B, C, D) -> Com_1(evalfbb4in(A, B, C, D)) :|: TRUE evalfbb1in(A, B, C, D) -> Com_1(evalfbb2in(A, B, C, D - 1)) :|: TRUE evalfbb4in(A, B, C, D) -> Com_1(evalfbb5in(D - C + 1, C - 1, C, D)) :|: 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(?, 32*Ar_1 + 14) 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(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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)) (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) (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(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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)) (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) (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(evalfbb5in) = 2 Pol(evalfbbin) = 2 Pol(evalfreturnin) = 1 Pol(evalfbb2in) = 2 Pol(evalfbb4in) = 2 Pol(evalfbb3in) = 2 Pol(evalfbb1in) = 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)) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= 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(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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)) (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) (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 A polynomial rank function with Pol(evalfstart) = V_2 + 1 Pol(evalfentryin) = V_2 + 1 Pol(evalfbb5in) = V_2 + 1 Pol(evalfbbin) = V_2 Pol(evalfreturnin) = V_2 Pol(evalfbb2in) = V_3 + 1 Pol(evalfbb4in) = V_3 Pol(evalfbb3in) = V_3 + 1 Pol(evalfbb1in) = V_3 + 1 Pol(evalfstop) = V_2 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] strictly and produces the following problem: 4: 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(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 1, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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)) (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) (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 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: 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(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 1, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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)) (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) (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 A polynomial rank function with Pol(evalfbb4in) = 1 Pol(evalfbb5in) = 0 Pol(evalfbb3in) = 2 Pol(evalfbb1in) = 2 Pol(evalfbb2in) = 2 and size complexities 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 S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3))", 0-0) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3))", 0-1) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3))", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3))", 0-3) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-0) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-1) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-2) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-3) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]", 0-0) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]", 0-1) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]", 0-3) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]", 0-0) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]", 0-1) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]", 0-3) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-0) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-1) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-3) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-0) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-1) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-3) = ? S("evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1))", 0-0) = ? S("evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1))", 0-1) = ? S("evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1))", 0-2) = ? S("evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1))", 0-3) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ]", 0-0) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ]", 0-1) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ]", 0-2) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ]", 0-3) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ]", 0-0) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ]", 0-1) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ]", 0-2) = ? S("evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ]", 0-3) = ? S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_1 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 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 orients the transitions evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) weakly and the transitions evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] strictly and produces the following problem: 6: 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(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 1, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: 2*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)) (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) (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 6 to obtain the following invariants: For symbol evalfbb1in: X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ -X_3 + X_4 >= 0 /\ X_2 + X_4 - 3 >= 0 /\ -X_2 + X_4 + 1 >= 0 /\ X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ -X_2 + X_3 + 1 >= 0 /\ X_2 - 2 >= 0 For symbol evalfbb2in: X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ -X_2 + X_3 + 1 >= 0 /\ X_2 - 2 >= 0 For symbol evalfbb3in: X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ -X_3 + X_4 >= 0 /\ X_2 + X_4 - 3 >= 0 /\ -X_2 + X_4 + 1 >= 0 /\ X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ -X_2 + X_3 + 1 >= 0 /\ X_2 - 2 >= 0 For symbol evalfbb4in: X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ -X_2 + X_3 + 1 >= 0 /\ X_2 - 2 >= 0 For symbol evalfbbin: X_2 - 2 >= 0 For symbol evalfreturnin: -X_2 + 1 >= 0 This yielded 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_1 + 1 >= 0 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] (Comp: 2*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 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ E >= 1 ] (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_3 >= Ar_2 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_2 >= Ar_3 + 1 ] (Comp: Ar_1 + 1, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) [ Ar_1 - 2 >= 0 ] (Comp: 2, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (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) = 6*V_2 Pol(evalfstart) = 6*V_2 Pol(evalfreturnin) = 3*V_1 + 3*V_2 Pol(evalfstop) = 3*V_1 + 3*V_2 Pol(evalfbb4in) = 3*V_4 Pol(evalfbb5in) = 3*V_1 + 3*V_2 Pol(evalfbb1in) = 3*V_4 - 1 Pol(evalfbb2in) = 3*V_4 + 1 Pol(evalfbb3in) = 3*V_4 Pol(evalfbbin) = 3*V_1 + 3*V_2 - 2 Pol(evalfentryin) = 6*V_2 orients all transitions weakly and the transitions evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ E >= 1 ] evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ 0 >= E + 1 ] evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_3 >= Ar_2 ] evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] strictly and produces the following problem: 8: 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_1 + 1 >= 0 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_3 - Ar_2 + 1, Ar_2 - 1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] (Comp: 6*Ar_1, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] (Comp: 2*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 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 ] (Comp: 6*Ar_1, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ E >= 1 ] (Comp: 6*Ar_1, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ -Ar_1 + Ar_3 + 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ 0 >= E + 1 ] (Comp: 6*Ar_1, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_3 >= Ar_2 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ -Ar_1 + Ar_2 + 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_2 >= Ar_3 + 1 ] (Comp: Ar_1 + 1, Cost: 1) evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_1 - 1, Ar_0 + Ar_1 - 1)) [ Ar_1 - 2 >= 0 ] (Comp: 2, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalfbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 2 ] (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb5in(Ar_1, Ar_1, Ar_2, Ar_3)) (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 32*Ar_1 + 14 Time: 0.369 sec (SMT: 0.297 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 -> evalfbb5in : A'=B, [], cost: 1 2: evalfbb5in -> evalfbbin : [ B>=2 ], cost: 1 3: evalfbb5in -> evalfreturnin : [ 1>=B ], cost: 1 4: evalfbbin -> evalfbb2in : C'=-1+B, D'=-1+A+B, [], cost: 1 5: evalfbb2in -> evalfbb4in : [ C>=1+D ], cost: 1 6: evalfbb2in -> evalfbb3in : [ D>=C ], cost: 1 7: evalfbb3in -> evalfbb1in : [ 0>=1+free ], cost: 1 8: evalfbb3in -> evalfbb1in : [ free_1>=1 ], cost: 1 9: evalfbb3in -> evalfbb4in : [], cost: 1 10: evalfbb1in -> evalfbb2in : D'=-1+D, [], cost: 1 11: evalfbb4in -> evalfbb5in : A'=1-C+D, B'=-1+C, [], cost: 1 12: evalfreturnin -> evalfstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalfstart -> evalfentryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb5in : A'=B, [], cost: 1 2: evalfbb5in -> evalfbbin : [ B>=2 ], cost: 1 4: evalfbbin -> evalfbb2in : C'=-1+B, D'=-1+A+B, [], cost: 1 5: evalfbb2in -> evalfbb4in : [ C>=1+D ], cost: 1 6: evalfbb2in -> evalfbb3in : [ D>=C ], cost: 1 7: evalfbb3in -> evalfbb1in : [ 0>=1+free ], cost: 1 8: evalfbb3in -> evalfbb1in : [ free_1>=1 ], cost: 1 9: evalfbb3in -> evalfbb4in : [], cost: 1 10: evalfbb1in -> evalfbb2in : D'=-1+D, [], cost: 1 11: evalfbb4in -> evalfbb5in : A'=1-C+D, B'=-1+C, [], cost: 1 Simplified all rules, resulting in: Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb5in : A'=B, [], cost: 1 2: evalfbb5in -> evalfbbin : [ B>=2 ], cost: 1 4: evalfbbin -> evalfbb2in : C'=-1+B, D'=-1+A+B, [], cost: 1 5: evalfbb2in -> evalfbb4in : [ C>=1+D ], cost: 1 6: evalfbb2in -> evalfbb3in : [ D>=C ], cost: 1 8: evalfbb3in -> evalfbb1in : [], cost: 1 9: evalfbb3in -> evalfbb4in : [], cost: 1 10: evalfbb1in -> evalfbb2in : D'=-1+D, [], cost: 1 11: evalfbb4in -> evalfbb5in : A'=1-C+D, B'=-1+C, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 14: evalfbb5in -> evalfbb2in : C'=-1+B, D'=-1+A+B, [ B>=2 ], cost: 2 5: evalfbb2in -> evalfbb4in : [ C>=1+D ], cost: 1 6: evalfbb2in -> evalfbb3in : [ D>=C ], cost: 1 9: evalfbb3in -> evalfbb4in : [], cost: 1 15: evalfbb3in -> evalfbb2in : D'=-1+D, [], cost: 2 11: evalfbb4in -> evalfbb5in : A'=1-C+D, B'=-1+C, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 14: evalfbb5in -> evalfbb2in : C'=-1+B, D'=-1+A+B, [ B>=2 ], cost: 2 17: evalfbb2in -> evalfbb2in : D'=-1+D, [ D>=C ], cost: 3 18: evalfbb2in -> evalfbb5in : A'=1-C+D, B'=-1+C, [ C>=1+D ], cost: 2 19: evalfbb2in -> evalfbb5in : A'=1-C+D, B'=-1+C, [ D>=C ], cost: 3 Accelerating simple loops of location 4. Accelerating the following rules: 17: evalfbb2in -> evalfbb2in : D'=-1+D, [ D>=C ], cost: 3 Accelerated rule 17 with metering function 1-C+D, yielding the new rule 20. Removing the simple loops: 17. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 14: evalfbb5in -> evalfbb2in : C'=-1+B, D'=-1+A+B, [ B>=2 ], cost: 2 18: evalfbb2in -> evalfbb5in : A'=1-C+D, B'=-1+C, [ C>=1+D ], cost: 2 19: evalfbb2in -> evalfbb5in : A'=1-C+D, B'=-1+C, [ D>=C ], cost: 3 20: evalfbb2in -> evalfbb2in : D'=-1+C, [ D>=C ], cost: 3-3*C+3*D Chained accelerated rules (with incoming rules): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 14: evalfbb5in -> evalfbb2in : C'=-1+B, D'=-1+A+B, [ B>=2 ], cost: 2 21: evalfbb5in -> evalfbb2in : C'=-1+B, D'=-2+B, [ B>=2 && -1+A+B>=-1+B ], cost: 5+3*A 18: evalfbb2in -> evalfbb5in : A'=1-C+D, B'=-1+C, [ C>=1+D ], cost: 2 19: evalfbb2in -> evalfbb5in : A'=1-C+D, B'=-1+C, [ D>=C ], cost: 3 Eliminated locations (on tree-shaped paths): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 22: evalfbb5in -> evalfbb5in : A'=1+A, B'=-2+B, C'=-1+B, D'=-1+A+B, [ B>=2 && -1+B>=A+B ], cost: 4 23: evalfbb5in -> evalfbb5in : A'=1+A, B'=-2+B, C'=-1+B, D'=-1+A+B, [ B>=2 && -1+A+B>=-1+B ], cost: 5 24: evalfbb5in -> evalfbb5in : A'=0, B'=-2+B, C'=-1+B, D'=-2+B, [ B>=2 && -1+A+B>=-1+B ], cost: 7+3*A 25: evalfbb5in -> [11] : [ B>=2 && -1+A+B>=-1+B ], cost: 5+3*A Accelerating simple loops of location 2. Accelerating the following rules: 22: evalfbb5in -> evalfbb5in : A'=1+A, B'=-2+B, C'=-1+B, D'=-1+A+B, [ B>=2 && -1+B>=A+B ], cost: 4 23: evalfbb5in -> evalfbb5in : A'=1+A, B'=-2+B, C'=-1+B, D'=-1+A+B, [ B>=2 && -1+A+B>=-1+B ], cost: 5 24: evalfbb5in -> evalfbb5in : A'=0, B'=-2+B, C'=-1+B, D'=-2+B, [ B>=2 && -1+A+B>=-1+B ], cost: 7+3*A Found no metering function for rule 22. Accelerated rule 23 with metering function meter (where 2*meter==-1+B), yielding the new rule 26. Accelerated rule 24 with metering function meter_1 (where 2*meter_1==-1+B), yielding the new rule 27. During metering: Instantiating temporary variables by {meter==1} During metering: Instantiating temporary variables by {meter==1} Nested simple loops 22 (outer loop) and 27 (inner loop) with metering function -1-A, resulting in the new rules: 28. During metering: Instantiating temporary variables by {meter_1==1} Removing the simple loops: 22 23 24. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 25: evalfbb5in -> [11] : [ B>=2 && -1+A+B>=-1+B ], cost: 5+3*A 26: evalfbb5in -> evalfbb5in : A'=A+meter, B'=-2*meter+B, C'=1-2*meter+B, D'=A-meter+B, [ B>=2 && -1+A+B>=-1+B && 2*meter==-1+B && meter>=1 ], cost: 5*meter 27: evalfbb5in -> evalfbb5in : A'=0, B'=-2*meter_1+B, C'=1-2*meter_1+B, D'=-2*meter_1+B, [ B>=2 && -1+A+B>=-1+B && 2*meter_1==-1+B && meter_1>=1 ], cost: 7*meter_1 28: evalfbb5in -> evalfbb5in : A'=0, B'=2+2*meter_1*(1+A)+2*A+B, C'=3+2*meter_1*(1+A)+2*A+B, D'=2+2*meter_1*(1+A)+2*A+B, [ 1+A==0 && -2+B>=2 && 2*meter_1==-3+B && meter_1>=1 && -1-A>=1 ], cost: -4-7*meter_1*(1+A)-4*A Chained accelerated rules (with incoming rules): Start location: evalfstart 13: evalfstart -> evalfbb5in : A'=B, [], cost: 2 29: evalfstart -> evalfbb5in : A'=meter+B, B'=-2*meter+B, C'=1-2*meter+B, D'=-meter+2*B, [ B>=2 && 2*meter==-1+B && meter>=1 ], cost: 2+5*meter 30: evalfstart -> evalfbb5in : A'=0, B'=-2*meter_1+B, C'=1-2*meter_1+B, D'=-2*meter_1+B, [ B>=2 && 2*meter_1==-1+B && meter_1>=1 ], cost: 2+7*meter_1 25: evalfbb5in -> [11] : [ B>=2 && -1+A+B>=-1+B ], cost: 5+3*A Eliminated locations (on tree-shaped paths): Start location: evalfstart 31: evalfstart -> [11] : A'=B, [ B>=2 ], cost: 7+3*B 32: evalfstart -> [13] : [ B>=2 && 2*meter==-1+B && meter>=1 ], cost: 2+5*meter 33: evalfstart -> [13] : [ B>=2 && 2*meter_1==-1+B && meter_1>=1 ], cost: 2+7*meter_1 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalfstart 31: evalfstart -> [11] : A'=B, [ B>=2 ], cost: 7+3*B 32: evalfstart -> [13] : [ B>=2 && 2*meter==-1+B && meter>=1 ], cost: 2+5*meter 33: evalfstart -> [13] : [ B>=2 && 2*meter_1==-1+B && meter_1>=1 ], cost: 2+7*meter_1 Computing asymptotic complexity for rule 31 Solved the limit problem by the following transformations: Created initial limit problem: -1+B (+/+!), 7+3*B (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 7+3*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 32 Solved the limit problem by the following transformations: Created initial limit problem: -1+B (+/+!), 2+5*meter (+), -2*meter+B (+/+!), 2+2*meter-B (+/+!) [not solved] applying transformation rule (C) using substitution {B==1+2*meter} resulting limit problem: 1 (+/+!), 2*meter (+/+!), 2+5*meter (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 2*meter (+/+!), 2+5*meter (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {meter==n} resulting limit problem: [solved] Solved the limit problem by the following transformations: Created initial limit problem: -1+B (+/+!), 2+5*meter (+), -2*meter+B (+/+!), 2+2*meter-B (+/+!) [not solved] applying transformation rule (C) using substitution {B==1+2*meter} resulting limit problem: 1 (+/+!), 2*meter (+/+!), 2+5*meter (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 2*meter (+/+!), 2+5*meter (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {meter==n} resulting limit problem: [solved] Solution: meter / n B / 1+2*n Resulting cost 2+5*n has complexity: Poly(n^1) Computing asymptotic complexity for rule 33 Solved the limit problem by the following transformations: Created initial limit problem: 2+2*meter_1-B (+/+!), -2*meter_1+B (+/+!), -1+B (+/+!), 2+7*meter_1 (+) [not solved] applying transformation rule (C) using substitution {B==1+2*meter_1} resulting limit problem: 1 (+/+!), 2*meter_1 (+/+!), 2+7*meter_1 (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 2*meter_1 (+/+!), 2+7*meter_1 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {meter_1==n} resulting limit problem: [solved] Solved the limit problem by the following transformations: Created initial limit problem: 2+2*meter_1-B (+/+!), -2*meter_1+B (+/+!), -1+B (+/+!), 2+7*meter_1 (+) [not solved] applying transformation rule (C) using substitution {B==1+2*meter_1} resulting limit problem: 1 (+/+!), 2*meter_1 (+/+!), 2+7*meter_1 (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 2*meter_1 (+/+!), 2+7*meter_1 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {meter_1==n} resulting limit problem: [solved] Solution: meter_1 / n B / 1+2*n Resulting cost 2+7*n has 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: 7+3*n Rule cost: 7+3*B Rule guard: [ B>=2 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)