/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 116 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f12(3, T, 3, 1, 0, F, G, H, I, J, K, L, M, N, O, P, Q, R, S)) :|: TRUE f12(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f12(A, B, C, T, E + 1, F, G, H, I, J, K, L, M, N, O, P, Q, R, S)) :|: C >= E + 1 f24(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f24(A, B, C, D, E, F, G + 1, T, I, J, K, L, M, N, O, P, Q, R, S)) :|: F >= G + 1 f36(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f36(A, B, C, D, E, F, G, H, I, J + 1, T, L, M, N, O, P, Q, R, S)) :|: I >= J + 1 f36(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f46(A, B, C, D, E, F, G, H, I, J, K, K, K, N, O, P, Q, R, S)) :|: J >= I f24(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f36(A, B, C, D, E, F, G, H, A, 0, 1, L, M, H, H, T, Q, R, S)) :|: G >= F f12(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) -> Com_1(f24(A, B, C, D, E, A, 0, 1, I, J, K, L, M, N, O, P, D, D, T)) :|: E >= C The start-symbols are:[f0_19] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 19) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f12(3, t, 3, 1, 0, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18)) (Comp: ?, Cost: 1) f12(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f12(ar_0, ar_1, ar_2, t, ar_4 + 1, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18)) [ ar_2 >= ar_4 + 1 ] (Comp: ?, Cost: 1) f24(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f24(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6 + 1, t, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18)) [ ar_5 >= ar_6 + 1 ] (Comp: ?, Cost: 1) f36(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9 + 1, t, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18)) [ ar_8 >= ar_9 + 1 ] (Comp: ?, Cost: 1) f36(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f46(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_10, ar_10, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18)) [ ar_9 >= ar_8 ] (Comp: ?, Cost: 1) f24(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_0, 0, 1, ar_11, ar_12, ar_7, ar_7, t, ar_16, ar_17, ar_18)) [ ar_6 >= ar_5 ] (Comp: ?, Cost: 1) f12(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f24(ar_0, ar_1, ar_2, ar_3, ar_4, ar_0, 0, 1, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_3, ar_3, t)) [ ar_4 >= ar_2 ] (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9]. We thus obtain the following problem: 2: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] (Comp: ?, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] (Comp: ?, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] (Comp: ?, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] (Comp: ?, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] (Comp: ?, Cost: 1) f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(3, 3, 0, ar_5, ar_6, ar_8, ar_9)) start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] (Comp: ?, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] (Comp: ?, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] (Comp: ?, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] (Comp: ?, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(3, 3, 0, ar_5, ar_6, ar_8, ar_9)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f12) = 3 Pol(f24) = 2 Pol(f36) = 1 Pol(f46) = 0 orients all transitions weakly and the transitions f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] (Comp: 3, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] (Comp: ?, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] (Comp: ?, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(3, 3, 0, ar_5, ar_6, ar_8, ar_9)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f12) = V_1 Pol(f24) = V_1 Pol(f36) = V_6 - V_7 Pol(f46) = V_6 - V_7 orients all transitions weakly and the transition f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] (Comp: 3, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] (Comp: ?, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(3, 3, 0, ar_5, ar_6, ar_8, ar_9)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f12) = V_1 Pol(f24) = V_4 - V_5 Pol(f36) = V_4 - V_5 Pol(f46) = V_4 - V_5 orients all transitions weakly and the transition f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] (Comp: 3, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] (Comp: 3, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] (Comp: ?, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(3, 3, 0, ar_5, ar_6, ar_8, ar_9)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f12) = V_2 - V_3 Pol(f24) = V_2 - V_3 Pol(f36) = V_2 - V_3 Pol(f46) = V_2 - V_3 orients all transitions weakly and the transition f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_0, 0, ar_8, ar_9)) [ ar_4 >= ar_2 ] (Comp: 3, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_0, 0)) [ ar_6 >= ar_5 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f46(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9)) [ ar_9 >= ar_8 ] (Comp: 3, Cost: 1) f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f36(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9 + 1)) [ ar_8 >= ar_9 + 1 ] (Comp: 3, Cost: 1) f24(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f24(ar_0, ar_2, ar_4, ar_5, ar_6 + 1, ar_8, ar_9)) [ ar_5 >= ar_6 + 1 ] (Comp: 3, Cost: 1) f12(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(ar_0, ar_2, ar_4 + 1, ar_5, ar_6, ar_8, ar_9)) [ ar_2 >= ar_4 + 1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_2, ar_4, ar_5, ar_6, ar_8, ar_9) -> Com_1(f12(3, 3, 0, ar_5, ar_6, ar_8, ar_9)) start location: koat_start leaf cost: 0 Complexity upper bound 19 Time: 0.135 sec (SMT: 0.106 sec) ---------------------------------------- (2) BOUNDS(1, 1)