WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 61 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 22 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(2(4(x1))) -> 2(0(0(5(3(3(5(2(0(4(x1)))))))))) 4(4(2(4(2(x1))))) -> 2(0(5(2(1(4(0(2(0(1(x1)))))))))) 0(5(4(2(4(3(x1)))))) -> 5(1(5(5(3(5(3(0(0(0(x1)))))))))) 1(1(4(5(3(3(x1)))))) -> 1(3(1(1(3(0(1(2(2(1(x1)))))))))) 3(1(4(3(1(2(x1)))))) -> 0(0(1(1(4(2(3(0(0(3(x1)))))))))) 3(2(4(2(4(1(x1)))))) -> 0(2(1(1(1(5(3(1(3(3(x1)))))))))) 3(3(0(4(1(2(x1)))))) -> 3(5(1(2(0(2(0(5(3(1(x1)))))))))) 4(1(4(5(0(5(4(x1))))))) -> 4(1(5(3(1(0(5(3(1(0(x1)))))))))) 4(4(0(5(4(2(2(x1))))))) -> 4(0(4(3(4(4(4(5(4(1(x1)))))))))) 5(4(5(3(2(4(3(x1))))))) -> 2(5(5(5(0(4(5(0(1(4(x1)))))))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(2(4(x1))) -> 2(0(0(5(3(3(5(2(0(4(x1)))))))))) 4(4(2(4(2(x1))))) -> 2(0(5(2(1(4(0(2(0(1(x1)))))))))) 0(5(4(2(4(3(x1)))))) -> 5(1(5(5(3(5(3(0(0(0(x1)))))))))) 1(1(4(5(3(3(x1)))))) -> 1(3(1(1(3(0(1(2(2(1(x1)))))))))) 3(1(4(3(1(2(x1)))))) -> 0(0(1(1(4(2(3(0(0(3(x1)))))))))) 3(2(4(2(4(1(x1)))))) -> 0(2(1(1(1(5(3(1(3(3(x1)))))))))) 3(3(0(4(1(2(x1)))))) -> 3(5(1(2(0(2(0(5(3(1(x1)))))))))) 4(1(4(5(0(5(4(x1))))))) -> 4(1(5(3(1(0(5(3(1(0(x1)))))))))) 4(4(0(5(4(2(2(x1))))))) -> 4(0(4(3(4(4(4(5(4(1(x1)))))))))) 5(4(5(3(2(4(3(x1))))))) -> 2(5(5(5(0(4(5(0(1(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(2(4(x1))) -> 2(0(0(5(3(3(5(2(0(4(x1)))))))))) 4(4(2(4(2(x1))))) -> 2(0(5(2(1(4(0(2(0(1(x1)))))))))) 0(5(4(2(4(3(x1)))))) -> 5(1(5(5(3(5(3(0(0(0(x1)))))))))) 1(1(4(5(3(3(x1)))))) -> 1(3(1(1(3(0(1(2(2(1(x1)))))))))) 3(1(4(3(1(2(x1)))))) -> 0(0(1(1(4(2(3(0(0(3(x1)))))))))) 3(2(4(2(4(1(x1)))))) -> 0(2(1(1(1(5(3(1(3(3(x1)))))))))) 3(3(0(4(1(2(x1)))))) -> 3(5(1(2(0(2(0(5(3(1(x1)))))))))) 4(1(4(5(0(5(4(x1))))))) -> 4(1(5(3(1(0(5(3(1(0(x1)))))))))) 4(4(0(5(4(2(2(x1))))))) -> 4(0(4(3(4(4(4(5(4(1(x1)))))))))) 5(4(5(3(2(4(3(x1))))))) -> 2(5(5(5(0(4(5(0(1(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(2(4(x1))) -> 2(0(0(5(3(3(5(2(0(4(x1)))))))))) 4(4(2(4(2(x1))))) -> 2(0(5(2(1(4(0(2(0(1(x1)))))))))) 0(5(4(2(4(3(x1)))))) -> 5(1(5(5(3(5(3(0(0(0(x1)))))))))) 1(1(4(5(3(3(x1)))))) -> 1(3(1(1(3(0(1(2(2(1(x1)))))))))) 3(1(4(3(1(2(x1)))))) -> 0(0(1(1(4(2(3(0(0(3(x1)))))))))) 3(2(4(2(4(1(x1)))))) -> 0(2(1(1(1(5(3(1(3(3(x1)))))))))) 3(3(0(4(1(2(x1)))))) -> 3(5(1(2(0(2(0(5(3(1(x1)))))))))) 4(1(4(5(0(5(4(x1))))))) -> 4(1(5(3(1(0(5(3(1(0(x1)))))))))) 4(4(0(5(4(2(2(x1))))))) -> 4(0(4(3(4(4(4(5(4(1(x1)))))))))) 5(4(5(3(2(4(3(x1))))))) -> 2(5(5(5(0(4(5(0(1(4(x1)))))))))) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. The certificate found is represented by the following graph. "[69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161] {(69,70,[4_1|0, 0_1|0, 1_1|0, 3_1|0, 5_1|0, encArg_1|0, encode_4_1|0, encode_2_1|0, encode_0_1|0, encode_5_1|0, encode_3_1|0, encode_1_1|0]), (69,71,[2_1|1, 4_1|1, 0_1|1, 1_1|1, 3_1|1, 5_1|1]), (69,72,[2_1|2]), (69,81,[2_1|2]), (69,90,[4_1|2]), (69,99,[4_1|2]), (69,108,[5_1|2]), (69,117,[1_1|2]), (69,126,[0_1|2]), (69,135,[0_1|2]), (69,144,[3_1|2]), (69,153,[2_1|2]), (70,70,[2_1|0, cons_4_1|0, cons_0_1|0, cons_1_1|0, cons_3_1|0, cons_5_1|0]), (71,70,[encArg_1|1]), (71,71,[2_1|1, 4_1|1, 0_1|1, 1_1|1, 3_1|1, 5_1|1]), (71,72,[2_1|2]), (71,81,[2_1|2]), (71,90,[4_1|2]), (71,99,[4_1|2]), (71,108,[5_1|2]), (71,117,[1_1|2]), (71,126,[0_1|2]), (71,135,[0_1|2]), (71,144,[3_1|2]), (71,153,[2_1|2]), (72,73,[0_1|2]), (73,74,[0_1|2]), (74,75,[5_1|2]), (75,76,[3_1|2]), (76,77,[3_1|2]), (77,78,[5_1|2]), (78,79,[2_1|2]), (79,80,[0_1|2]), (80,71,[4_1|2]), (80,90,[4_1|2]), (80,99,[4_1|2]), (80,72,[2_1|2]), (80,81,[2_1|2]), (81,82,[0_1|2]), (82,83,[5_1|2]), (83,84,[2_1|2]), (84,85,[1_1|2]), (85,86,[4_1|2]), (86,87,[0_1|2]), (87,88,[2_1|2]), (88,89,[0_1|2]), (89,71,[1_1|2]), (89,72,[1_1|2]), (89,81,[1_1|2]), (89,153,[1_1|2]), (89,117,[1_1|2]), (90,91,[0_1|2]), (91,92,[4_1|2]), (92,93,[3_1|2]), (93,94,[4_1|2]), (94,95,[4_1|2]), (95,96,[4_1|2]), (96,97,[5_1|2]), (97,98,[4_1|2]), (97,99,[4_1|2]), (98,71,[1_1|2]), (98,72,[1_1|2]), (98,81,[1_1|2]), (98,153,[1_1|2]), (98,117,[1_1|2]), (99,100,[1_1|2]), (100,101,[5_1|2]), (101,102,[3_1|2]), (102,103,[1_1|2]), (103,104,[0_1|2]), (104,105,[5_1|2]), (105,106,[3_1|2]), (106,107,[1_1|2]), (107,71,[0_1|2]), (107,90,[0_1|2]), (107,99,[0_1|2]), (107,108,[5_1|2]), (108,109,[1_1|2]), (109,110,[5_1|2]), (110,111,[5_1|2]), (111,112,[3_1|2]), (112,113,[5_1|2]), (113,114,[3_1|2]), (114,115,[0_1|2]), (115,116,[0_1|2]), (116,71,[0_1|2]), (116,144,[0_1|2]), (116,108,[5_1|2]), (117,118,[3_1|2]), (118,119,[1_1|2]), (119,120,[1_1|2]), (120,121,[3_1|2]), (121,122,[0_1|2]), (122,123,[1_1|2]), (123,124,[2_1|2]), (124,125,[2_1|2]), (125,71,[1_1|2]), (125,144,[1_1|2]), (125,117,[1_1|2]), (126,127,[0_1|2]), (127,128,[1_1|2]), (128,129,[1_1|2]), (129,130,[4_1|2]), (130,131,[2_1|2]), (131,132,[3_1|2]), (132,133,[0_1|2]), (133,134,[0_1|2]), (134,71,[3_1|2]), (134,72,[3_1|2]), (134,81,[3_1|2]), (134,153,[3_1|2]), (134,126,[0_1|2]), (134,135,[0_1|2]), (134,144,[3_1|2]), (135,136,[2_1|2]), (136,137,[1_1|2]), (137,138,[1_1|2]), (138,139,[1_1|2]), (139,140,[5_1|2]), (140,141,[3_1|2]), (141,142,[1_1|2]), (142,143,[3_1|2]), (142,144,[3_1|2]), (143,71,[3_1|2]), (143,117,[3_1|2]), (143,100,[3_1|2]), (143,126,[0_1|2]), (143,135,[0_1|2]), (143,144,[3_1|2]), (144,145,[5_1|2]), (145,146,[1_1|2]), (146,147,[2_1|2]), (147,148,[0_1|2]), (148,149,[2_1|2]), (149,150,[0_1|2]), (150,151,[5_1|2]), (151,152,[3_1|2]), (151,126,[0_1|2]), (152,71,[1_1|2]), (152,72,[1_1|2]), (152,81,[1_1|2]), (152,153,[1_1|2]), (152,117,[1_1|2]), (153,154,[5_1|2]), (154,155,[5_1|2]), (155,156,[5_1|2]), (156,157,[0_1|2]), (157,158,[4_1|2]), (158,159,[5_1|2]), (159,160,[0_1|2]), (160,161,[1_1|2]), (161,71,[4_1|2]), (161,144,[4_1|2]), (161,72,[2_1|2]), (161,81,[2_1|2]), (161,90,[4_1|2]), (161,99,[4_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)