4.80/2.21 WORST_CASE(NON_POLY, ?) 4.80/2.22 proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat 4.80/2.22 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.80/2.22 4.80/2.22 4.80/2.22 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF). 4.80/2.22 4.80/2.22 (0) CpxIntTrs 4.80/2.22 (1) Loat Proof [FINISHED, 532 ms] 4.80/2.22 (2) BOUNDS(INF, INF) 4.80/2.22 4.80/2.22 4.80/2.22 ---------------------------------------- 4.80/2.22 4.80/2.22 (0) 4.80/2.22 Obligation: 4.80/2.22 Complexity Int TRS consisting of the following rules: 4.80/2.22 f21(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: 0 >= A 4.80/2.22 f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: TRUE 4.80/2.22 f43(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f46(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: TRUE 4.80/2.22 f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(A, R, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: A >= 1 4.80/2.22 f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(A, S, C, 0, R, R, R, H, I, J, K, L, M, N, O, P, Q)) :|: 0 >= A && 999 + C >= R 4.80/2.22 f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(1, S, C, 0, R, R, R, H, I, J, K, L, M, N, O, P, Q)) :|: 0 >= A && R >= C + 1000 4.80/2.22 f21(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f29(0, B, R, D, E, F, G, 0, R, R, K, L, M, N, O, P, Q)) :|: A >= 1 4.80/2.22 f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f21(1, B, C, D, E, F, G, H, I, J, K, R, K, N, O, P, Q)) :|: 0 >= K 4.80/2.22 f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f21(1, B, C, D, E, F, G, H, I, J, S, R, 0, 1, S, S, S)) :|: S >= 1 && K >= 1 4.80/2.22 f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(1, B, C, D, E, F, G, H, I, J, S, R, 0, 1, S, S, S)) :|: 0 >= S && K >= 1 4.80/2.22 4.80/2.22 The start-symbols are:[f0_17] 4.80/2.22 4.80/2.22 4.80/2.22 ---------------------------------------- 4.80/2.22 4.80/2.22 (1) Loat Proof (FINISHED) 4.80/2.22 4.80/2.22 4.80/2.22 ### Pre-processing the ITS problem ### 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Initial linear ITS problem 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 0: f21 -> f29 : [ 0>=A ], cost: 1 4.80/2.22 4.80/2.22 6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 1: f41 -> f41 : [], cost: 1 4.80/2.22 4.80/2.22 2: f43 -> f46 : [], cost: 1 4.80/2.22 4.80/2.22 3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1 4.80/2.22 4.80/2.22 5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1 4.80/2.22 4.80/2.22 7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1 4.80/2.22 4.80/2.22 8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Removed unreachable and leaf rules: 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 0: f21 -> f29 : [ 0>=A ], cost: 1 4.80/2.22 4.80/2.22 6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 1: f41 -> f41 : [], cost: 1 4.80/2.22 4.80/2.22 3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1 4.80/2.22 4.80/2.22 5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1 4.80/2.22 4.80/2.22 7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1 4.80/2.22 4.80/2.22 8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 ### Simplification by acceleration and chaining ### 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Accelerating simple loops of location 1. 4.80/2.22 4.80/2.22 Accelerating the following rules: 4.80/2.22 4.80/2.22 1: f41 -> f41 : [], cost: 1 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Accelerated rule 1 with NONTERM, yielding the new rule 10. 4.80/2.22 4.80/2.22 Removing the simple loops: 1. 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Accelerated all simple loops using metering functions (where possible): 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 0: f21 -> f29 : [ 0>=A ], cost: 1 4.80/2.22 4.80/2.22 6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 10: f41 -> [6] : [], cost: INF 4.80/2.22 4.80/2.22 3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1 4.80/2.22 4.80/2.22 5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1 4.80/2.22 4.80/2.22 7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1 4.80/2.22 4.80/2.22 8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Chained accelerated rules (with incoming rules): 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 0: f21 -> f29 : [ 0>=A ], cost: 1 4.80/2.22 4.80/2.22 6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1 4.80/2.22 4.80/2.22 5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1 4.80/2.22 4.80/2.22 11: f29 -> [6] : B'=free, [ A>=1 ], cost: INF 4.80/2.22 4.80/2.22 12: f29 -> [6] : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 13: f29 -> [6] : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: INF 4.80/2.22 4.80/2.22 7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1 4.80/2.22 4.80/2.22 8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: INF 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Removed unreachable locations (and leaf rules with constant cost): 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 0: f21 -> f29 : [ 0>=A ], cost: 1 4.80/2.22 4.80/2.22 6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1 4.80/2.22 4.80/2.22 11: f29 -> [6] : B'=free, [ A>=1 ], cost: INF 4.80/2.22 4.80/2.22 12: f29 -> [6] : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 13: f29 -> [6] : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: INF 4.80/2.22 4.80/2.22 7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1 4.80/2.22 4.80/2.22 8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1 4.80/2.22 4.80/2.22 14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: INF 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Eliminated locations (on tree-shaped paths): 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 11: f29 -> [6] : B'=free, [ A>=1 ], cost: INF 4.80/2.22 4.80/2.22 12: f29 -> [6] : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 13: f29 -> [6] : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: INF 4.80/2.22 4.80/2.22 14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: INF 4.80/2.22 4.80/2.22 15: f0 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K ], cost: 2 4.80/2.22 4.80/2.22 16: f0 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 2 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Eliminated locations (on tree-shaped paths): 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: INF 4.80/2.22 4.80/2.22 17: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && 999+free_5>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 18: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && free_3>=1000+free_5 ], cost: INF 4.80/2.22 4.80/2.22 19: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && 999+free_5>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 20: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && free_3>=1000+free_5 ], cost: INF 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 ### Computing asymptotic complexity ### 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Fully simplified ITS problem 4.80/2.22 4.80/2.22 Start location: f0 4.80/2.22 4.80/2.22 14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: INF 4.80/2.22 4.80/2.22 17: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && 999+free_5>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 18: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && free_3>=1000+free_5 ], cost: INF 4.80/2.22 4.80/2.22 19: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && 999+free_5>=free_1 ], cost: INF 4.80/2.22 4.80/2.22 20: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && free_3>=1000+free_5 ], cost: INF 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Computing asymptotic complexity for rule 14 4.80/2.22 4.80/2.22 Resulting cost INF has complexity: Nonterm 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Found new complexity Nonterm. 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 Obtained the following overall complexity (w.r.t. the length of the input n): 4.80/2.22 4.80/2.22 Complexity: Nonterm 4.80/2.22 4.80/2.22 Cpx degree: Nonterm 4.80/2.22 4.80/2.22 Solved cost: INF 4.80/2.22 4.80/2.22 Rule cost: INF 4.80/2.22 4.80/2.22 Rule guard: [ 0>=free_10 && K>=1 ] 4.80/2.22 4.80/2.22 4.80/2.22 4.80/2.22 NO 4.80/2.22 4.80/2.22 4.80/2.22 ---------------------------------------- 4.80/2.22 4.80/2.22 (2) 4.80/2.22 BOUNDS(INF, INF) 4.80/2.25 EOF