/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 15 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f12(A, B, C, D, E) -> Com_1(f12(A + 1, B, C, D, E)) :|: 9 >= A f25(A, B, C, D, E) -> Com_1(f25(A, B + 1, C, D, E)) :|: 9 >= B f25(A, B, C, D, E) -> Com_1(f36(A, B, C, D, E)) :|: B >= 10 f12(A, B, C, D, E) -> Com_1(f25(A, 0, F, D, E)) :|: A >= 10 f0(A, B, C, D, E) -> Com_1(f12(0, B, C, F, G)) :|: TRUE The start-symbols are:[f0_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 25) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_0 ] (Comp: ?, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_1 ] (Comp: ?, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f36(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 10 ] (Comp: ?, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, 0, Fresh_2, Ar_3, Ar_4)) [ Ar_0 >= 10 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(0, Ar_1, Ar_2, Fresh_0, Fresh_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: ?, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_0 ] (Comp: ?, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_1 ] (Comp: ?, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f36(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 10 ] (Comp: ?, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, 0, Fresh_2, Ar_3, Ar_4)) [ Ar_0 >= 10 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(0, Ar_1, Ar_2, Fresh_0, Fresh_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f12) = 2 Pol(f25) = 1 Pol(f36) = 0 Pol(f0) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f36(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 10 ] f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, 0, Fresh_2, Ar_3, Ar_4)) [ Ar_0 >= 10 ] strictly and produces the following problem: 3: T: (Comp: ?, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_0 ] (Comp: ?, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_1 ] (Comp: 2, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f36(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 10 ] (Comp: 2, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, 0, Fresh_2, Ar_3, Ar_4)) [ Ar_0 >= 10 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(0, Ar_1, Ar_2, Fresh_0, Fresh_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f12) = -V_1 + 10 Pol(f25) = -V_1 Pol(f36) = -V_1 Pol(f0) = 10 Pol(koat_start) = 10 orients all transitions weakly and the transition f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 10, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_0 ] (Comp: ?, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_1 ] (Comp: 2, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f36(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 10 ] (Comp: 2, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, 0, Fresh_2, Ar_3, Ar_4)) [ Ar_0 >= 10 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(0, Ar_1, Ar_2, Fresh_0, Fresh_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f12) = 10 Pol(f25) = -V_2 + 10 Pol(f36) = -V_2 Pol(f0) = 10 Pol(koat_start) = 10 orients all transitions weakly and the transition f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_1 ] strictly and produces the following problem: 5: T: (Comp: 10, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_0 ] (Comp: 10, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ 9 >= Ar_1 ] (Comp: 2, Cost: 1) f25(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f36(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 10 ] (Comp: 2, Cost: 1) f12(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f25(Ar_0, 0, Fresh_2, Ar_3, Ar_4)) [ Ar_0 >= 10 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f12(0, Ar_1, Ar_2, Fresh_0, Fresh_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 25 Time: 0.077 sec (SMT: 0.065 sec) ---------------------------------------- (2) BOUNDS(1, 1)