/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).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, 118 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(Fresh_34, 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, Fresh_17 + Fresh_18, Fresh_19, Fresh_20 + Fresh_21, Fresh_22, Fresh_23 + Fresh_24, Fresh_25, Fresh_26 + Fresh_27, Fresh_28, Fresh_17 + Fresh_18 + Fresh_26 + Fresh_27, Fresh_17 + Fresh_18 - Fresh_26 - Fresh_27, Fresh_20 + Fresh_21 + Fresh_23 + Fresh_24, Fresh_20 + Fresh_21 - Fresh_23 - Fresh_24, Fresh_29, Fresh_30, Fresh_31 + Fresh_32, Fresh_33 + Fresh_32, Fresh_32)) [ 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, Fresh_0 + Fresh_1, Fresh_2, Fresh_3 + Fresh_4, Fresh_5, Fresh_6 + Fresh_7, Fresh_8, Fresh_9 + Fresh_10, Fresh_11, Fresh_0 + Fresh_1 + Fresh_9 + Fresh_10, Fresh_0 + Fresh_1 - Fresh_9 - Fresh_10, Fresh_3 + Fresh_4 + Fresh_6 + Fresh_7, Fresh_3 + Fresh_4 - Fresh_6 - Fresh_7, Fresh_12, Fresh_13, Fresh_14 + Fresh_15, Fresh_16 + Fresh_15, Fresh_15)) [ 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: 0.090 sec (SMT: 0.072 sec) ---------------------------------------- (2) BOUNDS(1, 1)