/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 41 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 38 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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 4. The certificate found is represented by the following graph. 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(21,202,[1_1|2]), (22,23,[1_1|1]), (23,24,[2_1|1]), (24,25,[3_1|1]), (25,26,[4_1|1]), (26,27,[5_1|1]), (27,28,[0_1|1]), (27,15,[0_1|1]), (27,33,[1_1|1]), (27,57,[1_1|1]), (28,29,[1_1|1]), (29,30,[2_1|1]), (30,31,[3_1|1]), (31,32,[4_1|1]), (32,6,[5_1|1]), (33,34,[2_1|1]), (34,35,[3_1|1]), (35,36,[4_1|1]), (36,37,[5_1|1]), (37,38,[1_1|1]), (38,39,[1_1|1]), (39,40,[0_1|1]), (39,232,[0_1|2]), (39,250,[1_1|2]), (39,274,[1_1|2]), (40,41,[1_1|1]), (41,42,[2_1|1]), (42,43,[3_1|1]), (43,44,[4_1|1]), (44,45,[5_1|1]), (45,46,[0_1|1]), (45,160,[0_1|2]), (45,178,[1_1|2]), (45,202,[1_1|2]), (46,47,[1_1|1]), (47,48,[2_1|1]), (48,49,[3_1|1]), (49,50,[4_1|1]), (50,51,[5_1|1]), (51,52,[0_1|1]), (51,15,[0_1|1]), (51,33,[1_1|1]), (51,57,[1_1|1]), (52,53,[1_1|1]), (53,54,[2_1|1]), (54,55,[3_1|1]), (55,56,[4_1|1]), (56,6,[5_1|1]), (57,58,[2_1|1]), (58,59,[3_1|1]), (59,60,[4_1|1]), (60,61,[5_1|1]), (61,62,[1_1|1]), (62,63,[1_1|1]), (63,64,[0_1|1]), (63,376,[0_1|2]), (63,394,[1_1|2]), (63,418,[1_1|2]), (64,65,[1_1|1]), (65,66,[2_1|1]), (66,67,[3_1|1]), (67,68,[4_1|1]), (68,69,[5_1|1]), (69,70,[0_1|1]), (69,232,[0_1|2]), (69,250,[1_1|2]), (69,274,[1_1|2]), (70,71,[1_1|1]), (71,72,[2_1|1]), (72,73,[3_1|1]), (73,74,[4_1|1]), (74,75,[5_1|1]), (75,76,[0_1|1]), (75,160,[0_1|2]), (75,178,[1_1|2]), (75,202,[1_1|2]), (76,77,[1_1|1]), (77,78,[2_1|1]), (78,79,[3_1|1]), (79,80,[4_1|1]), (80,81,[5_1|1]), (81,82,[0_1|1]), (81,15,[0_1|1]), (81,33,[1_1|1]), (81,57,[1_1|1]), (82,83,[1_1|1]), (83,84,[2_1|1]), (84,85,[3_1|1]), (85,86,[4_1|1]), (86,6,[5_1|1]), (87,6,[encArg_1|1]), (87,87,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1]), (87,88,[0_1|2]), (87,106,[1_1|2]), (87,130,[1_1|2]), (88,89,[2_1|2]), (89,90,[3_1|2]), (90,91,[4_1|2]), (91,92,[5_1|2]), (92,93,[1_1|2]), (93,94,[1_1|2]), (94,95,[0_1|2]), (94,304,[0_1|3]), (94,322,[1_1|3]), (94,346,[1_1|3]), (95,96,[1_1|2]), (96,97,[2_1|2]), (97,98,[3_1|2]), (98,99,[4_1|2]), (99,100,[5_1|2]), (100,101,[0_1|2]), (100,88,[0_1|2]), (100,106,[1_1|2]), 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(129,135,[5_1|2]), (130,131,[2_1|2]), (131,132,[3_1|2]), (132,133,[4_1|2]), (133,134,[5_1|2]), (134,135,[1_1|2]), (135,136,[1_1|2]), (136,137,[0_1|2]), (136,304,[0_1|3]), (136,322,[1_1|3]), (136,346,[1_1|3]), (137,138,[1_1|2]), (138,139,[2_1|2]), (139,140,[3_1|2]), (140,141,[4_1|2]), (141,142,[5_1|2]), (142,143,[0_1|2]), (142,304,[0_1|3]), (142,322,[1_1|3]), (142,346,[1_1|3]), (143,144,[1_1|2]), (144,145,[2_1|2]), (145,146,[3_1|2]), (146,147,[4_1|2]), (147,148,[5_1|2]), (148,149,[0_1|2]), (148,304,[0_1|3]), (148,322,[1_1|3]), (148,346,[1_1|3]), (149,150,[1_1|2]), (150,151,[2_1|2]), (151,152,[3_1|2]), (152,153,[4_1|2]), (153,154,[5_1|2]), (154,155,[0_1|2]), (154,88,[0_1|2]), (154,106,[1_1|2]), (154,130,[1_1|2]), (154,304,[0_1|3]), (154,322,[1_1|3]), (154,346,[1_1|3]), (155,156,[1_1|2]), (156,157,[2_1|2]), (157,158,[3_1|2]), (158,159,[4_1|2]), (159,87,[5_1|2]), (159,106,[5_1|2]), (159,130,[5_1|2]), (159,111,[5_1|2]), (159,135,[5_1|2]), (160,161,[2_1|2]), (161,162,[3_1|2]), (162,163,[4_1|2]), (163,164,[5_1|2]), (164,165,[1_1|2]), (165,166,[1_1|2]), (166,167,[0_1|2]), (167,168,[1_1|2]), (168,169,[2_1|2]), (169,170,[3_1|2]), (170,171,[4_1|2]), (171,172,[5_1|2]), (172,173,[0_1|2]), (173,174,[1_1|2]), (174,175,[2_1|2]), (175,176,[3_1|2]), (176,177,[4_1|2]), (177,33,[5_1|2]), (177,57,[5_1|2]), (178,179,[2_1|2]), (179,180,[3_1|2]), (180,181,[4_1|2]), (181,182,[5_1|2]), (182,183,[1_1|2]), (183,184,[1_1|2]), (184,185,[0_1|2]), (185,186,[1_1|2]), (186,187,[2_1|2]), (187,188,[3_1|2]), (188,189,[4_1|2]), (189,190,[5_1|2]), (190,191,[0_1|2]), (191,192,[1_1|2]), (192,193,[2_1|2]), (193,194,[3_1|2]), (194,195,[4_1|2]), (195,196,[5_1|2]), (196,197,[0_1|2]), (197,198,[1_1|2]), (198,199,[2_1|2]), (199,200,[3_1|2]), (200,201,[4_1|2]), (201,33,[5_1|2]), (201,57,[5_1|2]), (202,203,[2_1|2]), (203,204,[3_1|2]), (204,205,[4_1|2]), (205,206,[5_1|2]), (206,207,[1_1|2]), (207,208,[1_1|2]), (208,209,[0_1|2]), (209,210,[1_1|2]), (210,211,[2_1|2]), (211,212,[3_1|2]), (212,213,[4_1|2]), (213,214,[5_1|2]), (214,215,[0_1|2]), (215,216,[1_1|2]), (216,217,[2_1|2]), (217,218,[3_1|2]), (218,219,[4_1|2]), (219,220,[5_1|2]), (220,221,[0_1|2]), (221,222,[1_1|2]), (222,223,[2_1|2]), (223,224,[3_1|2]), (224,225,[4_1|2]), (225,226,[5_1|2]), (226,227,[0_1|2]), (227,228,[1_1|2]), (228,229,[2_1|2]), (229,230,[3_1|2]), (230,231,[4_1|2]), (231,33,[5_1|2]), (231,57,[5_1|2]), (232,233,[2_1|2]), (233,234,[3_1|2]), (234,235,[4_1|2]), (235,236,[5_1|2]), (236,237,[1_1|2]), (237,238,[1_1|2]), (238,239,[0_1|2]), (239,240,[1_1|2]), (240,241,[2_1|2]), (241,242,[3_1|2]), (242,243,[4_1|2]), (243,244,[5_1|2]), (244,245,[0_1|2]), (245,246,[1_1|2]), (246,247,[2_1|2]), (247,248,[3_1|2]), (248,249,[4_1|2]), (249,178,[5_1|2]), (249,202,[5_1|2]), (250,251,[2_1|2]), (251,252,[3_1|2]), (252,253,[4_1|2]), (253,254,[5_1|2]), (254,255,[1_1|2]), (255,256,[1_1|2]), (256,257,[0_1|2]), (257,258,[1_1|2]), (258,259,[2_1|2]), (259,260,[3_1|2]), (260,261,[4_1|2]), (261,262,[5_1|2]), (262,263,[0_1|2]), (263,264,[1_1|2]), (264,265,[2_1|2]), (265,266,[3_1|2]), (266,267,[4_1|2]), (267,268,[5_1|2]), (268,269,[0_1|2]), (269,270,[1_1|2]), (270,271,[2_1|2]), (271,272,[3_1|2]), (272,273,[4_1|2]), (273,178,[5_1|2]), (273,202,[5_1|2]), (274,275,[2_1|2]), (275,276,[3_1|2]), (276,277,[4_1|2]), (277,278,[5_1|2]), (278,279,[1_1|2]), (279,280,[1_1|2]), (280,281,[0_1|2]), (281,282,[1_1|2]), (282,283,[2_1|2]), (283,284,[3_1|2]), (284,285,[4_1|2]), (285,286,[5_1|2]), (286,287,[0_1|2]), (287,288,[1_1|2]), (288,289,[2_1|2]), (289,290,[3_1|2]), (290,291,[4_1|2]), (291,292,[5_1|2]), (292,293,[0_1|2]), (293,294,[1_1|2]), (294,295,[2_1|2]), (295,296,[3_1|2]), (296,297,[4_1|2]), (297,298,[5_1|2]), (298,299,[0_1|2]), (299,300,[1_1|2]), (300,301,[2_1|2]), (301,302,[3_1|2]), (302,303,[4_1|2]), (303,178,[5_1|2]), (303,202,[5_1|2]), (304,305,[2_1|3]), (305,306,[3_1|3]), (306,307,[4_1|3]), (307,308,[5_1|3]), (308,309,[1_1|3]), (309,310,[1_1|3]), (310,311,[0_1|3]), (310,520,[0_1|4]), (310,538,[1_1|4]), (310,562,[1_1|4]), (311,312,[1_1|3]), (312,313,[2_1|3]), (313,314,[3_1|3]), (314,315,[4_1|3]), (315,316,[5_1|3]), (316,317,[0_1|3]), (316,448,[0_1|4]), (316,466,[1_1|4]), (316,490,[1_1|4]), (317,318,[1_1|3]), (318,319,[2_1|3]), (319,320,[3_1|3]), (320,321,[4_1|3]), (321,106,[5_1|3]), (321,130,[5_1|3]), (321,322,[5_1|3]), (321,346,[5_1|3]), (321,112,[5_1|3]), (321,136,[5_1|3]), (322,323,[2_1|3]), (323,324,[3_1|3]), (324,325,[4_1|3]), (325,326,[5_1|3]), (326,327,[1_1|3]), (327,328,[1_1|3]), (328,329,[0_1|3]), (328,592,[0_1|4]), (328,610,[1_1|4]), (328,634,[1_1|4]), (329,330,[1_1|3]), (330,331,[2_1|3]), (331,332,[3_1|3]), (332,333,[4_1|3]), (333,334,[5_1|3]), (334,335,[0_1|3]), (334,520,[0_1|4]), (334,538,[1_1|4]), (334,562,[1_1|4]), (335,336,[1_1|3]), (336,337,[2_1|3]), (337,338,[3_1|3]), (338,339,[4_1|3]), (339,340,[5_1|3]), (340,341,[0_1|3]), (340,448,[0_1|4]), (340,466,[1_1|4]), (340,490,[1_1|4]), (341,342,[1_1|3]), (342,343,[2_1|3]), (343,344,[3_1|3]), (344,345,[4_1|3]), (345,106,[5_1|3]), (345,130,[5_1|3]), (345,322,[5_1|3]), (345,346,[5_1|3]), (345,112,[5_1|3]), (345,136,[5_1|3]), (346,347,[2_1|3]), (347,348,[3_1|3]), (348,349,[4_1|3]), (349,350,[5_1|3]), (350,351,[1_1|3]), (351,352,[1_1|3]), (352,353,[0_1|3]), (352,664,[0_1|4]), (352,682,[1_1|4]), (352,706,[1_1|4]), (353,354,[1_1|3]), (354,355,[2_1|3]), (355,356,[3_1|3]), (356,357,[4_1|3]), (357,358,[5_1|3]), (358,359,[0_1|3]), (358,592,[0_1|4]), (358,610,[1_1|4]), (358,634,[1_1|4]), (359,360,[1_1|3]), (360,361,[2_1|3]), (361,362,[3_1|3]), (362,363,[4_1|3]), (363,364,[5_1|3]), (364,365,[0_1|3]), (364,520,[0_1|4]), (364,538,[1_1|4]), (364,562,[1_1|4]), (365,366,[1_1|3]), (366,367,[2_1|3]), (367,368,[3_1|3]), (368,369,[4_1|3]), (369,370,[5_1|3]), (370,371,[0_1|3]), (370,448,[0_1|4]), (370,466,[1_1|4]), (370,490,[1_1|4]), (371,372,[1_1|3]), (372,373,[2_1|3]), (373,374,[3_1|3]), (374,375,[4_1|3]), (375,106,[5_1|3]), (375,130,[5_1|3]), (375,322,[5_1|3]), (375,346,[5_1|3]), (375,112,[5_1|3]), (375,136,[5_1|3]), (376,377,[2_1|2]), (377,378,[3_1|2]), (378,379,[4_1|2]), (379,380,[5_1|2]), (380,381,[1_1|2]), (381,382,[1_1|2]), (382,383,[0_1|2]), (383,384,[1_1|2]), (384,385,[2_1|2]), (385,386,[3_1|2]), (386,387,[4_1|2]), (387,388,[5_1|2]), (388,389,[0_1|2]), (389,390,[1_1|2]), (390,391,[2_1|2]), (391,392,[3_1|2]), (392,393,[4_1|2]), (393,250,[5_1|2]), (393,274,[5_1|2]), (394,395,[2_1|2]), (395,396,[3_1|2]), (396,397,[4_1|2]), (397,398,[5_1|2]), (398,399,[1_1|2]), (399,400,[1_1|2]), (400,401,[0_1|2]), (401,402,[1_1|2]), (402,403,[2_1|2]), (403,404,[3_1|2]), (404,405,[4_1|2]), (405,406,[5_1|2]), (406,407,[0_1|2]), (407,408,[1_1|2]), (408,409,[2_1|2]), (409,410,[3_1|2]), (410,411,[4_1|2]), (411,412,[5_1|2]), (412,413,[0_1|2]), (413,414,[1_1|2]), (414,415,[2_1|2]), (415,416,[3_1|2]), (416,417,[4_1|2]), (417,250,[5_1|2]), (417,274,[5_1|2]), (418,419,[2_1|2]), (419,420,[3_1|2]), (420,421,[4_1|2]), (421,422,[5_1|2]), (422,423,[1_1|2]), (423,424,[1_1|2]), (424,425,[0_1|2]), (425,426,[1_1|2]), (426,427,[2_1|2]), (427,428,[3_1|2]), (428,429,[4_1|2]), (429,430,[5_1|2]), (430,431,[0_1|2]), (431,432,[1_1|2]), (432,433,[2_1|2]), (433,434,[3_1|2]), (434,435,[4_1|2]), (435,436,[5_1|2]), (436,437,[0_1|2]), (437,438,[1_1|2]), (438,439,[2_1|2]), (439,440,[3_1|2]), (440,441,[4_1|2]), (441,442,[5_1|2]), (442,443,[0_1|2]), (443,444,[1_1|2]), (444,445,[2_1|2]), (445,446,[3_1|2]), (446,447,[4_1|2]), (447,250,[5_1|2]), (447,274,[5_1|2]), (448,449,[2_1|4]), (449,450,[3_1|4]), (450,451,[4_1|4]), (451,452,[5_1|4]), (452,453,[1_1|4]), (453,454,[1_1|4]), (454,455,[0_1|4]), (455,456,[1_1|4]), (456,457,[2_1|4]), (457,458,[3_1|4]), (458,459,[4_1|4]), (459,460,[5_1|4]), (460,461,[0_1|4]), (461,462,[1_1|4]), (462,463,[2_1|4]), (463,464,[3_1|4]), (464,465,[4_1|4]), (465,322,[5_1|4]), (465,346,[5_1|4]), (466,467,[2_1|4]), (467,468,[3_1|4]), (468,469,[4_1|4]), (469,470,[5_1|4]), (470,471,[1_1|4]), (471,472,[1_1|4]), (472,473,[0_1|4]), (473,474,[1_1|4]), (474,475,[2_1|4]), (475,476,[3_1|4]), (476,477,[4_1|4]), (477,478,[5_1|4]), (478,479,[0_1|4]), (479,480,[1_1|4]), (480,481,[2_1|4]), (481,482,[3_1|4]), (482,483,[4_1|4]), (483,484,[5_1|4]), (484,485,[0_1|4]), (485,486,[1_1|4]), (486,487,[2_1|4]), (487,488,[3_1|4]), (488,489,[4_1|4]), (489,322,[5_1|4]), (489,346,[5_1|4]), (490,491,[2_1|4]), (491,492,[3_1|4]), (492,493,[4_1|4]), (493,494,[5_1|4]), (494,495,[1_1|4]), (495,496,[1_1|4]), (496,497,[0_1|4]), (497,498,[1_1|4]), (498,499,[2_1|4]), (499,500,[3_1|4]), (500,501,[4_1|4]), (501,502,[5_1|4]), (502,503,[0_1|4]), (503,504,[1_1|4]), (504,505,[2_1|4]), (505,506,[3_1|4]), (506,507,[4_1|4]), (507,508,[5_1|4]), (508,509,[0_1|4]), (509,510,[1_1|4]), (510,511,[2_1|4]), (511,512,[3_1|4]), (512,513,[4_1|4]), (513,514,[5_1|4]), (514,515,[0_1|4]), (515,516,[1_1|4]), (516,517,[2_1|4]), (517,518,[3_1|4]), (518,519,[4_1|4]), (519,322,[5_1|4]), (519,346,[5_1|4]), (520,521,[2_1|4]), (521,522,[3_1|4]), (522,523,[4_1|4]), (523,524,[5_1|4]), (524,525,[1_1|4]), (525,526,[1_1|4]), (526,527,[0_1|4]), (527,528,[1_1|4]), (528,529,[2_1|4]), (529,530,[3_1|4]), (530,531,[4_1|4]), (531,532,[5_1|4]), (532,533,[0_1|4]), (533,534,[1_1|4]), (534,535,[2_1|4]), (535,536,[3_1|4]), (536,537,[4_1|4]), (537,466,[5_1|4]), (537,490,[5_1|4]), (538,539,[2_1|4]), (539,540,[3_1|4]), (540,541,[4_1|4]), (541,542,[5_1|4]), (542,543,[1_1|4]), (543,544,[1_1|4]), (544,545,[0_1|4]), (545,546,[1_1|4]), (546,547,[2_1|4]), (547,548,[3_1|4]), (548,549,[4_1|4]), (549,550,[5_1|4]), (550,551,[0_1|4]), (551,552,[1_1|4]), (552,553,[2_1|4]), (553,554,[3_1|4]), (554,555,[4_1|4]), (555,556,[5_1|4]), (556,557,[0_1|4]), (557,558,[1_1|4]), (558,559,[2_1|4]), (559,560,[3_1|4]), (560,561,[4_1|4]), (561,466,[5_1|4]), (561,490,[5_1|4]), (562,563,[2_1|4]), (563,564,[3_1|4]), (564,565,[4_1|4]), (565,566,[5_1|4]), (566,567,[1_1|4]), (567,568,[1_1|4]), (568,569,[0_1|4]), (569,570,[1_1|4]), (570,571,[2_1|4]), (571,572,[3_1|4]), (572,573,[4_1|4]), (573,574,[5_1|4]), (574,575,[0_1|4]), (575,576,[1_1|4]), (576,577,[2_1|4]), (577,578,[3_1|4]), (578,579,[4_1|4]), (579,580,[5_1|4]), (580,581,[0_1|4]), (581,582,[1_1|4]), (582,583,[2_1|4]), (583,584,[3_1|4]), (584,585,[4_1|4]), (585,586,[5_1|4]), (586,587,[0_1|4]), (587,588,[1_1|4]), (588,589,[2_1|4]), (589,590,[3_1|4]), (590,591,[4_1|4]), (591,466,[5_1|4]), (591,490,[5_1|4]), (592,593,[2_1|4]), (593,594,[3_1|4]), (594,595,[4_1|4]), (595,596,[5_1|4]), (596,597,[1_1|4]), (597,598,[1_1|4]), (598,599,[0_1|4]), (599,600,[1_1|4]), (600,601,[2_1|4]), (601,602,[3_1|4]), (602,603,[4_1|4]), (603,604,[5_1|4]), (604,605,[0_1|4]), (605,606,[1_1|4]), (606,607,[2_1|4]), (607,608,[3_1|4]), (608,609,[4_1|4]), (609,538,[5_1|4]), (609,562,[5_1|4]), (610,611,[2_1|4]), (611,612,[3_1|4]), (612,613,[4_1|4]), (613,614,[5_1|4]), (614,615,[1_1|4]), (615,616,[1_1|4]), (616,617,[0_1|4]), (617,618,[1_1|4]), (618,619,[2_1|4]), (619,620,[3_1|4]), (620,621,[4_1|4]), (621,622,[5_1|4]), (622,623,[0_1|4]), (623,624,[1_1|4]), (624,625,[2_1|4]), (625,626,[3_1|4]), (626,627,[4_1|4]), (627,628,[5_1|4]), (628,629,[0_1|4]), (629,630,[1_1|4]), (630,631,[2_1|4]), (631,632,[3_1|4]), (632,633,[4_1|4]), (633,538,[5_1|4]), (633,562,[5_1|4]), (634,635,[2_1|4]), (635,636,[3_1|4]), (636,637,[4_1|4]), (637,638,[5_1|4]), (638,639,[1_1|4]), (639,640,[1_1|4]), (640,641,[0_1|4]), (641,642,[1_1|4]), (642,643,[2_1|4]), (643,644,[3_1|4]), (644,645,[4_1|4]), (645,646,[5_1|4]), (646,647,[0_1|4]), (647,648,[1_1|4]), (648,649,[2_1|4]), (649,650,[3_1|4]), (650,651,[4_1|4]), (651,652,[5_1|4]), (652,653,[0_1|4]), (653,654,[1_1|4]), (654,655,[2_1|4]), (655,656,[3_1|4]), (656,657,[4_1|4]), (657,658,[5_1|4]), (658,659,[0_1|4]), (659,660,[1_1|4]), (660,661,[2_1|4]), (661,662,[3_1|4]), (662,663,[4_1|4]), (663,538,[5_1|4]), (663,562,[5_1|4]), (664,665,[2_1|4]), (665,666,[3_1|4]), (666,667,[4_1|4]), (667,668,[5_1|4]), (668,669,[1_1|4]), (669,670,[1_1|4]), (670,671,[0_1|4]), (671,672,[1_1|4]), (672,673,[2_1|4]), (673,674,[3_1|4]), (674,675,[4_1|4]), (675,676,[5_1|4]), (676,677,[0_1|4]), (677,678,[1_1|4]), (678,679,[2_1|4]), (679,680,[3_1|4]), (680,681,[4_1|4]), (681,610,[5_1|4]), (681,634,[5_1|4]), (682,683,[2_1|4]), (683,684,[3_1|4]), (684,685,[4_1|4]), (685,686,[5_1|4]), (686,687,[1_1|4]), (687,688,[1_1|4]), (688,689,[0_1|4]), (689,690,[1_1|4]), (690,691,[2_1|4]), (691,692,[3_1|4]), (692,693,[4_1|4]), (693,694,[5_1|4]), (694,695,[0_1|4]), (695,696,[1_1|4]), (696,697,[2_1|4]), (697,698,[3_1|4]), (698,699,[4_1|4]), (699,700,[5_1|4]), (700,701,[0_1|4]), (701,702,[1_1|4]), (702,703,[2_1|4]), (703,704,[3_1|4]), (704,705,[4_1|4]), (705,610,[5_1|4]), (705,634,[5_1|4]), (706,707,[2_1|4]), (707,708,[3_1|4]), (708,709,[4_1|4]), (709,710,[5_1|4]), (710,711,[1_1|4]), (711,712,[1_1|4]), (712,713,[0_1|4]), (713,714,[1_1|4]), (714,715,[2_1|4]), (715,716,[3_1|4]), (716,717,[4_1|4]), (717,718,[5_1|4]), (718,719,[0_1|4]), (719,720,[1_1|4]), (720,721,[2_1|4]), (721,722,[3_1|4]), (722,723,[4_1|4]), (723,724,[5_1|4]), (724,725,[0_1|4]), (725,726,[1_1|4]), (726,727,[2_1|4]), (727,728,[3_1|4]), (728,729,[4_1|4]), (729,730,[5_1|4]), (730,731,[0_1|4]), (731,732,[1_1|4]), (732,733,[2_1|4]), (733,734,[3_1|4]), (734,735,[4_1|4]), (735,610,[5_1|4]), (735,634,[5_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(2(3(4(5(1(x1))))))) ->^+ 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]. The pumping substitution is [x1 / 1(x1)]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL