5.11/2.41 WORST_CASE(NON_POLY, ?) 5.11/2.42 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.11/2.42 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.11/2.42 5.11/2.42 5.11/2.42 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF). 5.11/2.42 5.11/2.42 (0) CpxIntTrs 5.11/2.42 (1) Loat Proof [FINISHED, 724 ms] 5.11/2.42 (2) BOUNDS(INF, INF) 5.11/2.42 5.11/2.42 5.11/2.42 ---------------------------------------- 5.11/2.42 5.11/2.42 (0) 5.11/2.42 Obligation: 5.11/2.42 Complexity Int TRS consisting of the following rules: 5.11/2.42 f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f11(A, P, B, O, E, F, G, H, I, J, K, L, M, N)) :|: A >= 0 && B >= 1 5.11/2.42 f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f11(A, P, B, O, E, F, G, H, I, J, K, L, M, N)) :|: A >= 0 && 0 >= B + 1 5.11/2.42 f16(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f16(A, B, C, D, E, F, G + 1, O, O, O, K, L, M, N)) :|: E >= 0 && F >= G + 2 5.11/2.42 f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f13(A, 0, C, D, E, F, G, H, I, J, O, L, M, N)) :|: A >= 0 && B >= 0 && B <= 0 5.11/2.42 f16(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f11(A, R, P, Q, E, F, G, H, I, J, O, J, J, N)) :|: 1 + G >= F && P >= 1 && E >= 0 5.11/2.42 f16(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f11(A, R, P, Q, E, F, G, H, I, J, O, J, J, N)) :|: 1 + G >= F && 0 >= P + 1 && E >= 0 5.11/2.42 f300(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f16(A, B, C, D, E, F, 1, O, O, O, K, L, M, P)) :|: F >= 2 5.11/2.42 f300(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f13(A, 0, C, D, E, F, 0, H, I, 0, O, 0, 0, P)) :|: 1 >= F 5.11/2.42 5.11/2.42 The start-symbols are:[f300_14] 5.11/2.42 5.11/2.42 5.11/2.42 ---------------------------------------- 5.11/2.42 5.11/2.42 (1) Loat Proof (FINISHED) 5.11/2.42 5.11/2.42 5.11/2.42 ### Pre-processing the ITS problem ### 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Initial linear ITS problem 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 0: f11 -> f11 : B'=free_1, C'=B, D'=free, [ A>=0 && B>=1 ], cost: 1 5.11/2.42 5.11/2.42 1: f11 -> f11 : B'=free_3, C'=B, D'=free_2, [ A>=0 && 0>=1+B ], cost: 1 5.11/2.42 5.11/2.42 3: f11 -> f13 : B'=0, K'=free_5, [ A>=0 && B==0 ], cost: 1 5.11/2.42 5.11/2.42 2: f16 -> f16 : G'=1+G, H'=free_4, Q'=free_4, J'=free_4, [ E>=0 && F>=2+G ], cost: 1 5.11/2.42 5.11/2.42 4: f16 -> f11 : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 5: f16 -> f11 : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 6: f300 -> f16 : G'=1, H'=free_15, Q'=free_15, J'=free_15, N'=free_14, [ F>=2 ], cost: 1 5.11/2.42 5.11/2.42 7: f300 -> f13 : B'=0, G'=0, J'=0, K'=free_17, L'=0, M'=0, N'=free_16, [ 1>=F ], cost: 1 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Removed unreachable and leaf rules: 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 0: f11 -> f11 : B'=free_1, C'=B, D'=free, [ A>=0 && B>=1 ], cost: 1 5.11/2.42 5.11/2.42 1: f11 -> f11 : B'=free_3, C'=B, D'=free_2, [ A>=0 && 0>=1+B ], cost: 1 5.11/2.42 5.11/2.42 2: f16 -> f16 : G'=1+G, H'=free_4, Q'=free_4, J'=free_4, [ E>=0 && F>=2+G ], cost: 1 5.11/2.42 5.11/2.42 4: f16 -> f11 : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 5: f16 -> f11 : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 6: f300 -> f16 : G'=1, H'=free_15, Q'=free_15, J'=free_15, N'=free_14, [ F>=2 ], cost: 1 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 ### Simplification by acceleration and chaining ### 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Accelerating simple loops of location 0. 5.11/2.42 5.11/2.42 Accelerating the following rules: 5.11/2.42 5.11/2.42 0: f11 -> f11 : B'=free_1, C'=B, D'=free, [ A>=0 && B>=1 ], cost: 1 5.11/2.42 5.11/2.42 1: f11 -> f11 : B'=free_3, C'=B, D'=free_2, [ A>=0 && 0>=1+B ], cost: 1 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Accelerated rule 0 with NONTERM (after strengthening guard), yielding the new rule 8. 5.11/2.42 5.11/2.42 Accelerated rule 1 with NONTERM (after strengthening guard), yielding the new rule 9. 5.11/2.42 5.11/2.42 Removing the simple loops:. 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Accelerating simple loops of location 1. 5.11/2.42 5.11/2.42 Accelerating the following rules: 5.11/2.42 5.11/2.42 2: f16 -> f16 : G'=1+G, H'=free_4, Q'=free_4, J'=free_4, [ E>=0 && F>=2+G ], cost: 1 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Accelerated rule 2 with metering function -1+F-G, yielding the new rule 10. 5.11/2.42 5.11/2.42 Removing the simple loops: 2. 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Accelerated all simple loops using metering functions (where possible): 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 0: f11 -> f11 : B'=free_1, C'=B, D'=free, [ A>=0 && B>=1 ], cost: 1 5.11/2.42 5.11/2.42 1: f11 -> f11 : B'=free_3, C'=B, D'=free_2, [ A>=0 && 0>=1+B ], cost: 1 5.11/2.42 5.11/2.42 8: f11 -> [4] : [ A>=0 && B>=1 && free_1>=1 ], cost: INF 5.11/2.42 5.11/2.42 9: f11 -> [4] : [ A>=0 && 0>=1+B && 0>=1+free_3 ], cost: INF 5.11/2.42 5.11/2.42 4: f16 -> f11 : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 5: f16 -> f11 : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 10: f16 -> f16 : G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, [ E>=0 && F>=2+G ], cost: -1+F-G 5.11/2.42 5.11/2.42 6: f300 -> f16 : G'=1, H'=free_15, Q'=free_15, J'=free_15, N'=free_14, [ F>=2 ], cost: 1 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Chained accelerated rules (with incoming rules): 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 4: f16 -> f11 : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 5: f16 -> f11 : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 ], cost: 1 5.11/2.42 5.11/2.42 11: f16 -> f11 : B'=free_1, C'=free_9, D'=free, K'=free_8, L'=J, M'=J, [ 1+G>=F && E>=0 && A>=0 && free_9>=1 ], cost: 2 5.11/2.42 5.11/2.42 12: f16 -> f11 : B'=free_1, C'=free_13, D'=free, K'=free_12, L'=J, M'=J, [ 1+G>=F && E>=0 && A>=0 && free_13>=1 ], cost: 2 5.11/2.42 5.11/2.42 13: f16 -> f11 : B'=free_3, C'=free_9, D'=free_2, K'=free_8, L'=J, M'=J, [ 1+G>=F && E>=0 && A>=0 && 0>=1+free_9 ], cost: 2 5.11/2.42 5.11/2.42 14: f16 -> f11 : B'=free_3, C'=free_13, D'=free_2, K'=free_12, L'=J, M'=J, [ 1+G>=F && E>=0 && A>=0 && 0>=1+free_13 ], cost: 2 5.11/2.42 5.11/2.42 15: f16 -> [4] : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 16: f16 -> [4] : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 17: f16 -> [4] : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 && A>=0 && 0>=1+free_9 ], cost: INF 5.11/2.42 5.11/2.42 18: f16 -> [4] : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 && A>=0 && 0>=1+free_13 ], cost: INF 5.11/2.42 5.11/2.42 6: f300 -> f16 : G'=1, H'=free_15, Q'=free_15, J'=free_15, N'=free_14, [ F>=2 ], cost: 1 5.11/2.42 5.11/2.42 19: f300 -> f16 : G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, N'=free_14, [ E>=0 && F>=3 ], cost: -1+F 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Removed unreachable locations (and leaf rules with constant cost): 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 15: f16 -> [4] : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 16: f16 -> [4] : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 17: f16 -> [4] : B'=free_9, C'=free_6, D'=free_7, K'=free_8, L'=J, M'=J, [ 1+G>=F && free_6>=1 && E>=0 && A>=0 && 0>=1+free_9 ], cost: INF 5.11/2.42 5.11/2.42 18: f16 -> [4] : B'=free_13, C'=free_10, D'=free_11, K'=free_12, L'=J, M'=J, [ 1+G>=F && 0>=1+free_10 && E>=0 && A>=0 && 0>=1+free_13 ], cost: INF 5.11/2.42 5.11/2.42 6: f300 -> f16 : G'=1, H'=free_15, Q'=free_15, J'=free_15, N'=free_14, [ F>=2 ], cost: 1 5.11/2.42 5.11/2.42 19: f300 -> f16 : G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, N'=free_14, [ E>=0 && F>=3 ], cost: -1+F 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Eliminated locations (on tree-shaped paths): 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 20: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_8, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && free_6>=1 && E>=0 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 21: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_12, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && 0>=1+free_10 && E>=0 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 22: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_8, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && free_6>=1 && E>=0 && A>=0 && 0>=1+free_9 ], cost: INF 5.11/2.42 5.11/2.42 23: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_12, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && 0>=1+free_10 && E>=0 && A>=0 && 0>=1+free_13 ], cost: INF 5.11/2.42 5.11/2.42 24: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_8, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && free_6>=1 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 25: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_12, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && 0>=1+free_10 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 26: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_8, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && free_6>=1 && A>=0 && 0>=1+free_9 ], cost: INF 5.11/2.42 5.11/2.42 27: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_12, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && 0>=1+free_10 && A>=0 && 0>=1+free_13 ], cost: INF 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Applied pruning (of leafs and parallel rules): 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 20: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_8, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && free_6>=1 && E>=0 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 21: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_12, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && 0>=1+free_10 && E>=0 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 23: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_12, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && 0>=1+free_10 && E>=0 && A>=0 && 0>=1+free_13 ], cost: INF 5.11/2.42 5.11/2.42 24: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_8, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && free_6>=1 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 25: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_12, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && 0>=1+free_10 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 ### Computing asymptotic complexity ### 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Fully simplified ITS problem 5.11/2.42 5.11/2.42 Start location: f300 5.11/2.42 5.11/2.42 20: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_8, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && free_6>=1 && E>=0 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 21: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_12, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && 0>=1+free_10 && E>=0 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 23: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=1, H'=free_15, Q'=free_15, J'=free_15, K'=free_12, L'=free_15, M'=free_15, N'=free_14, [ F>=2 && 2>=F && 0>=1+free_10 && E>=0 && A>=0 && 0>=1+free_13 ], cost: INF 5.11/2.42 5.11/2.42 24: f300 -> [4] : B'=free_9, C'=free_6, D'=free_7, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_8, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && free_6>=1 && A>=0 && free_9>=1 ], cost: INF 5.11/2.42 5.11/2.42 25: f300 -> [4] : B'=free_13, C'=free_10, D'=free_11, G'=-1+F, H'=free_4, Q'=free_4, J'=free_4, K'=free_12, L'=free_4, M'=free_4, N'=free_14, [ E>=0 && F>=3 && 0>=1+free_10 && A>=0 && free_13>=1 ], cost: INF 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Computing asymptotic complexity for rule 20 5.11/2.42 5.11/2.42 Resulting cost INF has complexity: Nonterm 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Found new complexity Nonterm. 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 Obtained the following overall complexity (w.r.t. the length of the input n): 5.11/2.42 5.11/2.42 Complexity: Nonterm 5.11/2.42 5.11/2.42 Cpx degree: Nonterm 5.11/2.42 5.11/2.42 Solved cost: INF 5.11/2.42 5.11/2.42 Rule cost: INF 5.11/2.42 5.11/2.42 Rule guard: [ F>=2 && 2>=F && free_6>=1 && E>=0 && A>=0 && free_9>=1 ] 5.11/2.42 5.11/2.42 5.11/2.42 5.11/2.42 NO 5.11/2.42 5.11/2.42 5.11/2.42 ---------------------------------------- 5.11/2.42 5.11/2.42 (2) 5.11/2.42 BOUNDS(INF, INF) 5.11/2.44 EOF