/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).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, 126 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, T) -> Com_1(f6(0, 0, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: TRUE f6(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f6(U, B + 1, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: 63 >= B f14(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f14(A, B, C - 1, U + V, W, X + Y, Z, A1 + B1, C1, D1 + E1, F1, U + V + D1 + E1, U + V - D1 - E1, X + Y + A1 + B1, X + Y - A1 - B1, G1, H1, I1 + J1, K1 + J1, J1)) :|: C >= 0 f57(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f57(A, B, C - 1, U + V, W, X + Y, Z, A1 + B1, C1, D1 + E1, F1, U + V + D1 + E1, U + V - D1 - E1, X + Y + A1 + B1, X + Y - A1 - B1, G1, H1, I1 + J1, K1 + J1, J1)) :|: C >= 0 f57(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f101(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: 0 >= C + 1 f14(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f57(A, B, 7, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: 0 >= C + 1 f6(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f14(A, B, 7, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: B >= 64 The start-symbols are:[f0_20] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 106) 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, ar_19) -> Com_1(f6(0, 0, 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, ar_19)) (Comp: ?, Cost: 1) f6(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, ar_19) -> Com_1(f6(u, ar_1 + 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, ar_19)) [ 63 >= ar_1 ] (Comp: ?, Cost: 1) f14(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, ar_19) -> Com_1(f14(ar_0, ar_1, ar_2 - 1, u + v, w, x + y, z, a1 + b1, c1, d1 + e1, f1, u + v + d1 + e1, u + v - d1 - e1, x + y + a1 + b1, x + y - a1 - b1, g1, h1, i1 + j1, k1 + j1, j1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f57(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, ar_19) -> Com_1(f57(ar_0, ar_1, ar_2 - 1, u + v, w, x + y, z, a1 + b1, c1, d1 + e1, f1, u + v + d1 + e1, u + v - d1 - e1, x + y + a1 + b1, x + y - a1 - b1, g1, h1, i1 + j1, k1 + j1, j1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f57(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, ar_19) -> Com_1(f101(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, ar_19)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f14(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, ar_19) -> Com_1(f57(ar_0, ar_1, 7, 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, ar_19)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f6(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, ar_19) -> Com_1(f14(ar_0, ar_1, 7, 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, ar_19)) [ ar_1 >= 64 ] (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, ar_19) -> 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, ar_19)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_1, ar_2]. We thus obtain the following problem: 2: T: (Comp: 1, Cost: 0) koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] (Comp: ?, Cost: 1) f0(ar_1, ar_2) -> Com_1(f6(0, ar_2)) 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_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] (Comp: 1, Cost: 1) f0(ar_1, ar_2) -> Com_1(f6(0, ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f6) = 3 Pol(f14) = 2 Pol(f57) = 1 Pol(f101) = 0 orients all transitions weakly and the transitions f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] (Comp: 3, Cost: 1) f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] (Comp: 3, Cost: 1) f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] (Comp: 1, Cost: 1) f0(ar_1, ar_2) -> Com_1(f6(0, ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 8 Pol(f0) = 8 Pol(f6) = 8 Pol(f14) = 8 Pol(f57) = V_2 + 1 Pol(f101) = V_2 orients all transitions weakly and the transition f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] (Comp: 3, Cost: 1) f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] (Comp: 3, Cost: 1) f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] (Comp: 8, Cost: 1) f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] (Comp: 1, Cost: 1) f0(ar_1, ar_2) -> Com_1(f6(0, ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 64 Pol(f0) = 64 Pol(f6) = -V_1 + 64 Pol(f14) = -V_1 Pol(f57) = -V_1 Pol(f101) = -V_1 orients all transitions weakly and the transition f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] (Comp: 3, Cost: 1) f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] (Comp: 3, Cost: 1) f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] (Comp: 8, Cost: 1) f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: 64, Cost: 1) f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] (Comp: 1, Cost: 1) f0(ar_1, ar_2) -> Com_1(f6(0, ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f14) = V_2 + 1 and size complexities S("f0(ar_1, ar_2) -> Com_1(f6(0, ar_2))", 0-0) = 0 S("f0(ar_1, ar_2) -> Com_1(f6(0, ar_2))", 0-1) = ar_2 S("f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ]", 0-0) = 64 S("f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ]", 0-1) = ar_2 S("f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ]", 0-0) = 64 S("f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ]", 0-1) = 7 S("f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ]", 0-0) = 64 S("f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ]", 0-1) = 7 S("f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ]", 0-0) = 64 S("f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ]", 0-1) = 7 S("f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ]", 0-0) = 64 S("f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ]", 0-1) = 7 S("f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ]", 0-0) = 64 S("f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ]", 0-1) = 7 S("koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_1 S("koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_2 orients the transitions f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] weakly and the transition f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(ar_1, ar_2) -> Com_1(f0(ar_1, ar_2)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f6(ar_1, ar_2) -> Com_1(f14(ar_1, 7)) [ ar_1 >= 64 ] (Comp: 3, Cost: 1) f14(ar_1, ar_2) -> Com_1(f57(ar_1, 7)) [ 0 >= ar_2 + 1 ] (Comp: 3, Cost: 1) f57(ar_1, ar_2) -> Com_1(f101(ar_1, ar_2)) [ 0 >= ar_2 + 1 ] (Comp: 8, Cost: 1) f57(ar_1, ar_2) -> Com_1(f57(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: 24, Cost: 1) f14(ar_1, ar_2) -> Com_1(f14(ar_1, ar_2 - 1)) [ ar_2 >= 0 ] (Comp: 64, Cost: 1) f6(ar_1, ar_2) -> Com_1(f6(ar_1 + 1, ar_2)) [ 63 >= ar_1 ] (Comp: 1, Cost: 1) f0(ar_1, ar_2) -> Com_1(f6(0, ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 106 Time: 0.107 sec (SMT: 0.095 sec) ---------------------------------------- (2) BOUNDS(1, 1)