/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 179 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: 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: 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(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 (innermost) 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: 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: 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: 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: 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: 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 3. The certificate found is represented by the following graph. "[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, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282] {(72,73,[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]), (72,74,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (72,75,[3_1|2]), (72,80,[0_1|2]), (72,85,[0_1|2]), (72,90,[0_1|2]), (72,95,[4_1|2]), (72,100,[2_1|2]), (72,105,[5_1|2]), (72,110,[1_1|2]), (72,115,[3_1|2]), (72,120,[4_1|2]), (72,125,[3_1|2]), (72,130,[2_1|2]), (72,135,[1_1|2]), (72,140,[0_1|2]), (72,145,[0_1|2]), (72,150,[4_1|2]), (72,155,[3_1|2]), (72,160,[3_1|2]), (72,165,[0_1|2]), (73,73,[cons_1_1|0, cons_5_1|0, cons_0_1|0, cons_3_1|0, cons_4_1|0, cons_2_1|0]), (74,73,[encArg_1|1]), (74,74,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (74,75,[3_1|2]), (74,80,[0_1|2]), (74,85,[0_1|2]), (74,90,[0_1|2]), (74,95,[4_1|2]), (74,100,[2_1|2]), (74,105,[5_1|2]), (74,110,[1_1|2]), (74,115,[3_1|2]), (74,120,[4_1|2]), (74,125,[3_1|2]), (74,130,[2_1|2]), (74,135,[1_1|2]), (74,140,[0_1|2]), (74,145,[0_1|2]), (74,150,[4_1|2]), (74,155,[3_1|2]), (74,160,[3_1|2]), (74,165,[0_1|2]), (75,76,[1_1|2]), (76,77,[1_1|2]), (77,78,[2_1|2]), (78,79,[2_1|2]), (79,74,[4_1|2]), (79,95,[4_1|2]), (79,120,[4_1|2]), (79,150,[4_1|2]), (79,155,[3_1|2]), (79,160,[3_1|2]), (80,81,[4_1|2]), (81,82,[5_1|2]), (82,83,[0_1|2]), (83,84,[2_1|2]), (84,74,[1_1|2]), (84,95,[1_1|2]), (84,120,[1_1|2]), (84,150,[1_1|2]), (84,75,[3_1|2]), (84,80,[0_1|2]), (84,85,[0_1|2]), (84,90,[0_1|2]), (84,248,[3_1|3]), (85,86,[0_1|2]), (86,87,[1_1|2]), (87,88,[3_1|2]), (88,89,[4_1|2]), (88,155,[3_1|2]), (88,253,[3_1|3]), (89,74,[1_1|2]), (89,105,[1_1|2]), (89,75,[3_1|2]), (89,80,[0_1|2]), (89,85,[0_1|2]), (89,90,[0_1|2]), (89,248,[3_1|3]), (90,91,[2_1|2]), (91,92,[5_1|2]), (92,93,[2_1|2]), (93,94,[0_1|2]), (94,74,[4_1|2]), (94,95,[4_1|2]), (94,120,[4_1|2]), (94,150,[4_1|2]), (94,155,[3_1|2]), (94,160,[3_1|2]), (95,96,[2_1|2]), (96,97,[3_1|2]), (97,98,[1_1|2]), (98,99,[1_1|2]), (99,74,[1_1|2]), (99,95,[1_1|2]), (99,120,[1_1|2]), (99,150,[1_1|2]), (99,75,[3_1|2]), (99,80,[0_1|2]), (99,85,[0_1|2]), (99,90,[0_1|2]), (99,248,[3_1|3]), (100,101,[4_1|2]), (101,102,[0_1|2]), (102,103,[4_1|2]), (103,104,[4_1|2]), (104,74,[0_1|2]), (104,80,[0_1|2]), (104,85,[0_1|2]), (104,90,[0_1|2]), (104,140,[0_1|2]), (104,145,[0_1|2]), (104,165,[0_1|2]), (104,130,[2_1|2]), (104,135,[1_1|2]), (104,258,[2_1|3]), (105,106,[1_1|2]), (106,107,[5_1|2]), (107,108,[2_1|2]), (108,109,[1_1|2]), (109,74,[0_1|2]), (109,80,[0_1|2]), (109,85,[0_1|2]), (109,90,[0_1|2]), (109,140,[0_1|2]), (109,145,[0_1|2]), (109,165,[0_1|2]), (109,130,[2_1|2]), (109,135,[1_1|2]), (109,258,[2_1|3]), (110,111,[0_1|2]), (111,112,[4_1|2]), (112,113,[0_1|2]), (113,114,[2_1|2]), (114,74,[2_1|2]), (114,80,[2_1|2]), (114,85,[2_1|2]), (114,90,[2_1|2]), (114,140,[2_1|2]), (114,145,[2_1|2]), (114,165,[2_1|2, 0_1|2]), (114,86,[2_1|2]), (115,116,[0_1|2]), (116,117,[4_1|2]), (117,118,[5_1|2]), (118,119,[0_1|2]), (119,74,[2_1|2]), (119,100,[2_1|2]), (119,130,[2_1|2]), (119,91,[2_1|2]), (119,141,[2_1|2]), (119,165,[0_1|2]), (120,121,[1_1|2]), (121,122,[1_1|2]), (122,123,[3_1|2]), (123,124,[2_1|2]), (124,74,[4_1|2]), (124,95,[4_1|2]), (124,120,[4_1|2]), (124,150,[4_1|2]), (124,155,[3_1|2]), (124,160,[3_1|2]), (125,126,[4_1|2]), (126,127,[4_1|2]), (127,128,[1_1|2]), (128,129,[2_1|2]), (129,74,[2_1|2]), (129,95,[2_1|2]), (129,120,[2_1|2]), (129,150,[2_1|2]), (129,165,[0_1|2]), (130,131,[1_1|2]), (131,132,[1_1|2]), (132,133,[0_1|2]), (133,134,[2_1|2]), (134,74,[0_1|2]), (134,80,[0_1|2]), (134,85,[0_1|2]), (134,90,[0_1|2]), (134,140,[0_1|2]), (134,145,[0_1|2]), (134,165,[0_1|2]), (134,116,[0_1|2]), (134,130,[2_1|2]), (134,135,[1_1|2]), (134,258,[2_1|3]), (135,136,[0_1|2]), (136,137,[1_1|2]), (137,138,[3_1|2]), (138,139,[4_1|2]), (139,74,[2_1|2]), (139,105,[2_1|2]), (139,165,[0_1|2]), (140,141,[2_1|2]), (141,142,[0_1|2]), (142,143,[0_1|2]), (142,263,[2_1|3]), (143,144,[3_1|2]), (144,74,[0_1|2]), (144,80,[0_1|2]), (144,85,[0_1|2]), (144,90,[0_1|2]), (144,140,[0_1|2]), (144,145,[0_1|2]), (144,165,[0_1|2]), (144,130,[2_1|2]), (144,135,[1_1|2]), (144,258,[2_1|3]), (145,146,[1_1|2]), (146,147,[3_1|2]), (147,148,[4_1|2]), (148,149,[3_1|2]), (149,74,[4_1|2]), (149,95,[4_1|2]), (149,120,[4_1|2]), (149,150,[4_1|2]), (149,155,[3_1|2]), (149,160,[3_1|2]), (150,151,[1_1|2]), (151,152,[3_1|2]), (152,153,[4_1|2]), (153,154,[2_1|2]), (154,74,[3_1|2]), (154,95,[3_1|2]), (154,120,[3_1|2]), (154,150,[3_1|2, 4_1|2]), (155,156,[3_1|2]), (156,157,[2_1|2]), (157,158,[2_1|2]), (158,159,[3_1|2]), (159,74,[1_1|2]), (159,95,[1_1|2]), (159,120,[1_1|2]), (159,150,[1_1|2]), (159,75,[3_1|2]), (159,80,[0_1|2]), (159,85,[0_1|2]), (159,90,[0_1|2]), (159,248,[3_1|3]), (160,161,[3_1|2]), (161,162,[2_1|2]), (162,163,[3_1|2]), (162,278,[4_1|3]), (163,164,[5_1|2]), (163,125,[3_1|2]), (163,268,[3_1|3]), (163,273,[4_1|3]), (164,74,[5_1|2]), (164,105,[5_1|2]), (164,95,[4_1|2]), (164,100,[2_1|2]), (164,110,[1_1|2]), (164,115,[3_1|2]), (164,120,[4_1|2]), (164,125,[3_1|2]), (164,273,[4_1|3]), (165,166,[4_1|2]), (166,167,[1_1|2]), (167,168,[2_1|2]), (168,169,[4_1|2]), (169,74,[0_1|2]), (169,80,[0_1|2]), (169,85,[0_1|2]), (169,90,[0_1|2]), (169,140,[0_1|2]), (169,145,[0_1|2]), (169,165,[0_1|2]), (169,130,[2_1|2]), (169,135,[1_1|2]), (169,258,[2_1|3]), (248,249,[1_1|3]), (249,250,[1_1|3]), (250,251,[2_1|3]), (251,252,[2_1|3]), (252,95,[4_1|3]), (252,120,[4_1|3]), (252,150,[4_1|3]), (253,254,[3_1|3]), (254,255,[2_1|3]), (255,256,[2_1|3]), (256,257,[3_1|3]), (257,95,[1_1|3]), (257,120,[1_1|3]), (257,150,[1_1|3]), (258,259,[1_1|3]), (259,260,[1_1|3]), (260,261,[0_1|3]), (261,262,[2_1|3]), (262,116,[0_1|3]), (263,264,[1_1|3]), (264,265,[1_1|3]), (265,266,[0_1|3]), (266,267,[2_1|3]), (267,74,[0_1|3]), (267,80,[0_1|3]), (267,85,[0_1|3]), (267,90,[0_1|3]), (267,140,[0_1|3, 0_1|2]), (267,145,[0_1|3, 0_1|2]), (267,165,[0_1|3]), (267,130,[2_1|2]), (267,135,[1_1|2]), (267,258,[2_1|3]), (268,269,[4_1|3]), (269,270,[4_1|3]), (270,271,[1_1|3]), (271,272,[2_1|3]), (272,95,[2_1|3]), (272,120,[2_1|3]), (272,150,[2_1|3]), (273,274,[2_1|3]), (274,275,[3_1|3]), (275,276,[1_1|3]), (276,277,[1_1|3]), (277,95,[1_1|3]), (277,120,[1_1|3]), (277,150,[1_1|3]), (277,273,[1_1|3]), (278,279,[1_1|3]), (279,280,[3_1|3]), (280,281,[4_1|3]), (281,282,[2_1|3]), (282,95,[3_1|3]), (282,120,[3_1|3]), (282,273,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)