/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 46 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 (full) 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: FULL ---------------------------------------- (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 (full) 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: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) 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: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) 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: FULL ---------------------------------------- (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. "[19, 20, 25, 26, 27, 28, 29, 30, 31, 32, 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] {(19,20,[b_1|0, encArg_1|0, encode_b_1|0, encode_a_1|0]), (19,25,[a_1|1, b_1|1]), (19,26,[a_1|2]), (20,20,[a_1|0, cons_b_1|0]), (25,20,[encArg_1|1]), (25,25,[a_1|1, b_1|1]), (25,26,[a_1|2]), (26,27,[a_1|2]), (27,28,[a_1|2]), (28,29,[b_1|2]), (28,44,[a_1|3]), (29,30,[a_1|2]), (30,31,[b_1|2]), (30,26,[a_1|2]), (30,37,[a_1|3]), (31,32,[a_1|2]), (32,25,[b_1|2]), (32,26,[b_1|2, a_1|2]), (32,27,[b_1|2]), (32,44,[a_1|3]), (37,38,[a_1|3]), (38,39,[a_1|3]), (39,40,[b_1|3]), (39,51,[a_1|4]), (40,41,[a_1|3]), (41,42,[b_1|3]), (41,37,[a_1|3]), (41,58,[a_1|4]), (42,43,[a_1|3]), (43,27,[b_1|3]), (43,28,[b_1|3]), (43,44,[a_1|3, b_1|3]), (44,45,[a_1|3]), (45,46,[a_1|3]), (46,47,[b_1|3]), (47,48,[a_1|3]), (48,49,[b_1|3]), (48,37,[a_1|3]), (48,65,[a_1|4]), (49,50,[a_1|3]), (50,26,[b_1|3]), (50,37,[b_1|3]), (50,44,[b_1|3]), (51,52,[a_1|4]), (52,53,[a_1|4]), (53,54,[b_1|4]), (54,55,[a_1|4]), (55,56,[b_1|4]), (55,58,[a_1|4]), (56,57,[a_1|4]), (57,44,[b_1|4]), (58,59,[a_1|4]), (59,60,[a_1|4]), (60,61,[b_1|4]), (60,86,[a_1|5]), (61,62,[a_1|4]), (62,63,[b_1|4]), (63,64,[a_1|4]), (64,45,[b_1|4]), (64,46,[b_1|4]), (64,72,[a_1|4]), (65,66,[a_1|4]), (66,67,[a_1|4]), (67,68,[b_1|4]), (68,69,[a_1|4]), (69,70,[b_1|4]), (69,79,[a_1|5]), (70,71,[a_1|4]), (71,39,[b_1|4]), (71,46,[b_1|4]), (72,73,[a_1|4]), (73,74,[a_1|4]), (74,75,[b_1|4]), (75,76,[a_1|4]), (76,77,[b_1|4]), (76,65,[a_1|4]), (76,93,[a_1|5]), (77,78,[a_1|4]), (78,37,[b_1|4]), (78,65,[b_1|4]), (79,80,[a_1|5]), (80,81,[a_1|5]), (81,82,[b_1|5]), (81,107,[a_1|6]), (82,83,[a_1|5]), (83,84,[b_1|5]), (84,85,[a_1|5]), (85,52,[b_1|5]), (85,100,[a_1|5]), (86,87,[a_1|5]), (87,88,[a_1|5]), (88,89,[b_1|5]), (89,90,[a_1|5]), (90,91,[b_1|5]), (90,114,[a_1|5]), (91,92,[a_1|5]), (92,72,[b_1|5]), (93,94,[a_1|5]), (94,95,[a_1|5]), (95,96,[b_1|5]), (96,97,[a_1|5]), (97,98,[b_1|5]), (98,99,[a_1|5]), (99,67,[b_1|5]), (100,101,[a_1|5]), (101,102,[a_1|5]), (102,103,[b_1|5]), (103,104,[a_1|5]), (104,105,[b_1|5]), (104,121,[a_1|5]), (105,106,[a_1|5]), (106,58,[b_1|5]), (107,108,[a_1|6]), (108,109,[a_1|6]), (109,110,[b_1|6]), (110,111,[a_1|6]), (111,112,[b_1|6]), (111,128,[a_1|6]), (112,113,[a_1|6]), (113,100,[b_1|6]), (114,115,[a_1|5]), (115,116,[a_1|5]), (116,117,[b_1|5]), (117,118,[a_1|5]), (118,119,[b_1|5]), (119,120,[a_1|5]), (120,74,[b_1|5]), (121,122,[a_1|5]), (122,123,[a_1|5]), (123,124,[b_1|5]), (124,125,[a_1|5]), (125,126,[b_1|5]), (125,135,[a_1|6]), (126,127,[a_1|5]), (127,60,[b_1|5]), (128,129,[a_1|6]), (129,130,[a_1|6]), (130,131,[b_1|6]), (131,132,[a_1|6]), (132,133,[b_1|6]), (133,134,[a_1|6]), (134,102,[b_1|6]), (135,136,[a_1|6]), (136,137,[a_1|6]), (137,138,[b_1|6]), (137,149,[a_1|7]), (138,139,[a_1|6]), (139,140,[b_1|6]), (140,141,[a_1|6]), (141,87,[b_1|6]), (141,142,[a_1|6]), (142,143,[a_1|6]), (143,144,[a_1|6]), (144,145,[b_1|6]), (145,146,[a_1|6]), (146,147,[b_1|6]), (146,156,[a_1|6]), (147,148,[a_1|6]), (148,114,[b_1|6]), (149,150,[a_1|7]), (150,151,[a_1|7]), (151,152,[b_1|7]), (152,153,[a_1|7]), (153,154,[b_1|7]), (153,163,[a_1|7]), (154,155,[a_1|7]), (155,142,[b_1|7]), (156,157,[a_1|6]), (157,158,[a_1|6]), (158,159,[b_1|6]), (159,160,[a_1|6]), (160,161,[b_1|6]), (161,162,[a_1|6]), (162,116,[b_1|6]), (163,164,[a_1|7]), (164,165,[a_1|7]), (165,166,[b_1|7]), (166,167,[a_1|7]), (167,168,[b_1|7]), (168,169,[a_1|7]), (169,144,[b_1|7])}" ---------------------------------------- (8) BOUNDS(1, n^1)