/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 52 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 30 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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(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(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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 3. The certificate found is represented by the following graph. "[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, 217, 219, 221, 222, 224, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270] {(73,74,[1_1|0, 5_1|0, 0_1|0, 3_1|0, 4_1|0, 2_1|0, encArg_1|0, encode_1_1|0, encode_4_1|0, encode_3_1|0, encode_2_1|0, encode_5_1|0, encode_0_1|0]), (73,75,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (73,76,[3_1|2]), (73,81,[0_1|2]), (73,86,[0_1|2]), (73,91,[0_1|2]), (73,96,[4_1|2]), (73,101,[2_1|2]), (73,106,[5_1|2]), (73,111,[1_1|2]), (73,116,[3_1|2]), (73,121,[4_1|2]), (73,126,[3_1|2]), (73,131,[2_1|2]), (73,136,[1_1|2]), (73,141,[0_1|2]), (73,146,[0_1|2]), (73,151,[4_1|2]), (73,156,[3_1|2]), (73,161,[3_1|2]), (73,166,[0_1|2]), (74,74,[cons_1_1|0, cons_5_1|0, cons_0_1|0, cons_3_1|0, cons_4_1|0, cons_2_1|0]), (75,74,[encArg_1|1]), (75,75,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (75,76,[3_1|2]), (75,81,[0_1|2]), (75,86,[0_1|2]), (75,91,[0_1|2]), (75,96,[4_1|2]), (75,101,[2_1|2]), (75,106,[5_1|2]), (75,111,[1_1|2]), (75,116,[3_1|2]), (75,121,[4_1|2]), (75,126,[3_1|2]), (75,131,[2_1|2]), (75,136,[1_1|2]), (75,141,[0_1|2]), (75,146,[0_1|2]), (75,151,[4_1|2]), (75,156,[3_1|2]), (75,161,[3_1|2]), (75,166,[0_1|2]), (76,77,[1_1|2]), (77,78,[1_1|2]), (78,79,[2_1|2]), (79,80,[2_1|2]), (80,75,[4_1|2]), (80,96,[4_1|2]), (80,121,[4_1|2]), (80,151,[4_1|2]), (80,156,[3_1|2]), (80,161,[3_1|2]), (81,82,[4_1|2]), (82,83,[5_1|2]), (83,84,[0_1|2]), (84,85,[2_1|2]), (85,75,[1_1|2]), (85,96,[1_1|2]), (85,121,[1_1|2]), (85,151,[1_1|2]), (85,76,[3_1|2]), (85,81,[0_1|2]), (85,86,[0_1|2]), (85,91,[0_1|2]), (85,217,[3_1|3]), (86,87,[0_1|2]), (87,88,[1_1|2]), (88,89,[3_1|2]), (89,90,[4_1|2]), (89,156,[3_1|2]), (89,228,[3_1|3]), (90,75,[1_1|2]), (90,106,[1_1|2]), (90,76,[3_1|2]), (90,81,[0_1|2]), (90,86,[0_1|2]), (90,91,[0_1|2]), (90,217,[3_1|3]), (91,92,[2_1|2]), (92,93,[5_1|2]), (93,94,[2_1|2]), (94,95,[0_1|2]), (95,75,[4_1|2]), (95,96,[4_1|2]), (95,121,[4_1|2]), (95,151,[4_1|2]), (95,156,[3_1|2]), (95,161,[3_1|2]), (96,97,[2_1|2]), (97,98,[3_1|2]), (98,99,[1_1|2]), (99,100,[1_1|2]), (100,75,[1_1|2]), (100,96,[1_1|2]), (100,121,[1_1|2]), (100,151,[1_1|2]), (100,76,[3_1|2]), (100,81,[0_1|2]), (100,86,[0_1|2]), (100,91,[0_1|2]), (100,217,[3_1|3]), (101,102,[4_1|2]), (102,103,[0_1|2]), (103,104,[4_1|2]), (104,105,[4_1|2]), (105,75,[0_1|2]), (105,81,[0_1|2]), (105,86,[0_1|2]), (105,91,[0_1|2]), (105,141,[0_1|2]), (105,146,[0_1|2]), (105,166,[0_1|2]), (105,131,[2_1|2]), (105,136,[1_1|2]), (105,233,[2_1|3]), (106,107,[1_1|2]), (107,108,[5_1|2]), (108,109,[2_1|2]), (109,110,[1_1|2]), (110,75,[0_1|2]), (110,81,[0_1|2]), (110,86,[0_1|2]), (110,91,[0_1|2]), (110,141,[0_1|2]), (110,146,[0_1|2]), (110,166,[0_1|2]), (110,131,[2_1|2]), (110,136,[1_1|2]), (110,233,[2_1|3]), (111,112,[0_1|2]), (112,113,[4_1|2]), (113,114,[0_1|2]), (114,115,[2_1|2]), (115,75,[2_1|2]), (115,81,[2_1|2]), (115,86,[2_1|2]), (115,91,[2_1|2]), (115,141,[2_1|2]), (115,146,[2_1|2]), (115,166,[2_1|2, 0_1|2]), (115,87,[2_1|2]), (116,117,[0_1|2]), (117,118,[4_1|2]), (118,119,[5_1|2]), (119,120,[0_1|2]), (120,75,[2_1|2]), (120,101,[2_1|2]), (120,131,[2_1|2]), (120,92,[2_1|2]), (120,142,[2_1|2]), (120,166,[0_1|2]), (121,122,[1_1|2]), (122,123,[1_1|2]), (123,124,[3_1|2]), (124,125,[2_1|2]), (125,75,[4_1|2]), (125,96,[4_1|2]), (125,121,[4_1|2]), (125,151,[4_1|2]), (125,156,[3_1|2]), (125,161,[3_1|2]), (126,127,[4_1|2]), (127,128,[4_1|2]), (128,129,[1_1|2]), (129,130,[2_1|2]), (130,75,[2_1|2]), (130,96,[2_1|2]), (130,121,[2_1|2]), (130,151,[2_1|2]), (130,166,[0_1|2]), (131,132,[1_1|2]), (132,133,[1_1|2]), (133,134,[0_1|2]), (134,135,[2_1|2]), (135,75,[0_1|2]), (135,81,[0_1|2]), (135,86,[0_1|2]), (135,91,[0_1|2]), (135,141,[0_1|2]), (135,146,[0_1|2]), (135,166,[0_1|2]), (135,117,[0_1|2]), (135,131,[2_1|2]), (135,136,[1_1|2]), (135,233,[2_1|3]), (136,137,[0_1|2]), (137,138,[1_1|2]), (138,139,[3_1|2]), (139,140,[4_1|2]), (140,75,[2_1|2]), (140,106,[2_1|2]), (140,166,[0_1|2]), (141,142,[2_1|2]), (142,143,[0_1|2]), (143,144,[0_1|2]), (143,251,[2_1|3]), (144,145,[3_1|2]), (145,75,[0_1|2]), (145,81,[0_1|2]), (145,86,[0_1|2]), (145,91,[0_1|2]), (145,141,[0_1|2]), (145,146,[0_1|2]), (145,166,[0_1|2]), (145,131,[2_1|2]), (145,136,[1_1|2]), (145,233,[2_1|3]), (146,147,[1_1|2]), (147,148,[3_1|2]), (148,149,[4_1|2]), (149,150,[3_1|2]), (150,75,[4_1|2]), (150,96,[4_1|2]), (150,121,[4_1|2]), (150,151,[4_1|2]), (150,156,[3_1|2]), (150,161,[3_1|2]), (151,152,[1_1|2]), (152,153,[3_1|2]), (153,154,[4_1|2]), (154,155,[2_1|2]), (155,75,[3_1|2]), (155,96,[3_1|2]), (155,121,[3_1|2]), (155,151,[3_1|2, 4_1|2]), (156,157,[3_1|2]), (157,158,[2_1|2]), (158,159,[2_1|2]), (159,160,[3_1|2]), (160,75,[1_1|2]), (160,96,[1_1|2]), (160,121,[1_1|2]), (160,151,[1_1|2]), (160,76,[3_1|2]), (160,81,[0_1|2]), (160,86,[0_1|2]), (160,91,[0_1|2]), (160,217,[3_1|3]), (161,162,[3_1|2]), (162,163,[2_1|2]), (163,164,[3_1|2]), (163,266,[4_1|3]), (164,165,[5_1|2]), (164,126,[3_1|2]), (164,256,[3_1|3]), (164,261,[4_1|3]), (165,75,[5_1|2]), (165,106,[5_1|2]), (165,96,[4_1|2]), (165,101,[2_1|2]), (165,111,[1_1|2]), (165,116,[3_1|2]), (165,121,[4_1|2]), (165,126,[3_1|2]), (165,261,[4_1|3]), (166,167,[4_1|2]), (167,168,[1_1|2]), (168,169,[2_1|2]), (169,170,[4_1|2]), (170,75,[0_1|2]), (170,81,[0_1|2]), (170,86,[0_1|2]), (170,91,[0_1|2]), (170,141,[0_1|2]), (170,146,[0_1|2]), (170,166,[0_1|2]), (170,131,[2_1|2]), (170,136,[1_1|2]), (170,233,[2_1|3]), (217,219,[1_1|3]), (219,221,[1_1|3]), (221,222,[2_1|3]), (222,224,[2_1|3]), (224,96,[4_1|3]), (224,121,[4_1|3]), (224,151,[4_1|3]), (228,229,[3_1|3]), (229,230,[2_1|3]), (230,231,[2_1|3]), (231,232,[3_1|3]), (232,96,[1_1|3]), (232,121,[1_1|3]), (232,151,[1_1|3]), (233,234,[1_1|3]), (234,235,[1_1|3]), (235,236,[0_1|3]), (236,237,[2_1|3]), (237,117,[0_1|3]), (251,252,[1_1|3]), (252,253,[1_1|3]), (253,254,[0_1|3]), (254,255,[2_1|3]), (255,75,[0_1|3]), (255,81,[0_1|3]), (255,86,[0_1|3]), (255,91,[0_1|3]), (255,141,[0_1|3, 0_1|2]), (255,146,[0_1|3, 0_1|2]), (255,166,[0_1|3]), (255,131,[2_1|2]), (255,136,[1_1|2]), (255,233,[2_1|3]), (256,257,[4_1|3]), (257,258,[4_1|3]), (258,259,[1_1|3]), (259,260,[2_1|3]), (260,96,[2_1|3]), (260,121,[2_1|3]), (260,151,[2_1|3]), (261,262,[2_1|3]), (262,263,[3_1|3]), (263,264,[1_1|3]), (264,265,[1_1|3]), (265,96,[1_1|3]), (265,121,[1_1|3]), (265,151,[1_1|3]), (265,261,[1_1|3]), (266,267,[1_1|3]), (267,268,[3_1|3]), (268,269,[4_1|3]), (269,270,[2_1|3]), (270,96,[3_1|3]), (270,121,[3_1|3]), (270,261,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)