/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 54 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 0 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: b(a(b(a(a(x1))))) -> a(a(a(b(a(b(a(b(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(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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: b(a(b(a(a(x1))))) -> a(a(a(b(a(b(a(b(x1)))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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: b(a(b(a(a(x1))))) -> a(a(a(b(a(b(a(b(x1)))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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: b(a(b(a(a(x1))))) -> a(a(a(b(a(b(a(b(x1)))))))) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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 7. The certificate found is represented by the following graph. "[29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 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, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171] {(29,30,[b_1|0, encArg_1|0, encode_b_1|0, encode_a_1|0]), (29,31,[a_1|1, b_1|1]), (29,32,[a_1|2]), (30,30,[a_1|0, cons_b_1|0]), (31,30,[encArg_1|1]), (31,31,[a_1|1, b_1|1]), (31,32,[a_1|2]), (32,33,[a_1|2]), (33,34,[a_1|2]), (34,35,[b_1|2]), (34,46,[a_1|3]), (35,36,[a_1|2]), (36,37,[b_1|2]), (36,32,[a_1|2]), (36,39,[a_1|3]), (37,38,[a_1|2]), (38,31,[b_1|2]), (38,32,[b_1|2, a_1|2]), (38,33,[b_1|2]), (38,46,[a_1|3]), (39,40,[a_1|3]), (40,41,[a_1|3]), (41,42,[b_1|3]), (41,53,[a_1|4]), (42,43,[a_1|3]), (43,44,[b_1|3]), (43,39,[a_1|3]), (43,60,[a_1|4]), (44,45,[a_1|3]), (45,33,[b_1|3]), (45,34,[b_1|3]), (45,46,[a_1|3, b_1|3]), (46,47,[a_1|3]), (47,48,[a_1|3]), (48,49,[b_1|3]), (49,50,[a_1|3]), (50,51,[b_1|3]), (50,39,[a_1|3]), (50,67,[a_1|4]), (51,52,[a_1|3]), (52,32,[b_1|3]), (52,39,[b_1|3]), (52,46,[b_1|3]), (53,54,[a_1|4]), (54,55,[a_1|4]), (55,56,[b_1|4]), (56,57,[a_1|4]), (57,58,[b_1|4]), (57,60,[a_1|4]), (58,59,[a_1|4]), (59,46,[b_1|4]), (60,61,[a_1|4]), (61,62,[a_1|4]), (62,63,[b_1|4]), (62,88,[a_1|5]), (63,64,[a_1|4]), (64,65,[b_1|4]), (65,66,[a_1|4]), (66,47,[b_1|4]), (66,48,[b_1|4]), (66,74,[a_1|4]), (67,68,[a_1|4]), (68,69,[a_1|4]), (69,70,[b_1|4]), (70,71,[a_1|4]), (71,72,[b_1|4]), (71,81,[a_1|5]), (72,73,[a_1|4]), (73,41,[b_1|4]), (73,48,[b_1|4]), (74,75,[a_1|4]), (75,76,[a_1|4]), (76,77,[b_1|4]), (77,78,[a_1|4]), (78,79,[b_1|4]), (78,67,[a_1|4]), (78,95,[a_1|5]), (79,80,[a_1|4]), (80,39,[b_1|4]), (80,67,[b_1|4]), (81,82,[a_1|5]), (82,83,[a_1|5]), (83,84,[b_1|5]), (83,109,[a_1|6]), (84,85,[a_1|5]), (85,86,[b_1|5]), (86,87,[a_1|5]), (87,54,[b_1|5]), (87,102,[a_1|5]), (88,89,[a_1|5]), (89,90,[a_1|5]), (90,91,[b_1|5]), (91,92,[a_1|5]), (92,93,[b_1|5]), (92,116,[a_1|5]), (93,94,[a_1|5]), (94,74,[b_1|5]), (95,96,[a_1|5]), (96,97,[a_1|5]), (97,98,[b_1|5]), (98,99,[a_1|5]), (99,100,[b_1|5]), (100,101,[a_1|5]), (101,69,[b_1|5]), (102,103,[a_1|5]), (103,104,[a_1|5]), (104,105,[b_1|5]), (105,106,[a_1|5]), (106,107,[b_1|5]), (106,123,[a_1|5]), (107,108,[a_1|5]), (108,60,[b_1|5]), (109,110,[a_1|6]), (110,111,[a_1|6]), (111,112,[b_1|6]), (112,113,[a_1|6]), (113,114,[b_1|6]), (113,130,[a_1|6]), (114,115,[a_1|6]), (115,102,[b_1|6]), (116,117,[a_1|5]), (117,118,[a_1|5]), (118,119,[b_1|5]), (119,120,[a_1|5]), (120,121,[b_1|5]), (121,122,[a_1|5]), (122,76,[b_1|5]), (123,124,[a_1|5]), (124,125,[a_1|5]), (125,126,[b_1|5]), (126,127,[a_1|5]), (127,128,[b_1|5]), (127,137,[a_1|6]), (128,129,[a_1|5]), (129,62,[b_1|5]), (130,131,[a_1|6]), (131,132,[a_1|6]), (132,133,[b_1|6]), (133,134,[a_1|6]), (134,135,[b_1|6]), (135,136,[a_1|6]), (136,104,[b_1|6]), (137,138,[a_1|6]), (138,139,[a_1|6]), (139,140,[b_1|6]), (139,151,[a_1|7]), (140,141,[a_1|6]), (141,142,[b_1|6]), (142,143,[a_1|6]), (143,89,[b_1|6]), (143,144,[a_1|6]), (144,145,[a_1|6]), (145,146,[a_1|6]), (146,147,[b_1|6]), (147,148,[a_1|6]), (148,149,[b_1|6]), (148,158,[a_1|6]), (149,150,[a_1|6]), (150,116,[b_1|6]), (151,152,[a_1|7]), (152,153,[a_1|7]), (153,154,[b_1|7]), (154,155,[a_1|7]), (155,156,[b_1|7]), (155,165,[a_1|7]), (156,157,[a_1|7]), (157,144,[b_1|7]), (158,159,[a_1|6]), (159,160,[a_1|6]), (160,161,[b_1|6]), (161,162,[a_1|6]), (162,163,[b_1|6]), (163,164,[a_1|6]), (164,118,[b_1|6]), (165,166,[a_1|7]), (166,167,[a_1|7]), (167,168,[b_1|7]), (168,169,[a_1|7]), (169,170,[b_1|7]), (170,171,[a_1|7]), (171,146,[b_1|7])}" ---------------------------------------- (8) BOUNDS(1, n^1)