/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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, 36 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 (innermost) 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: 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(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 (innermost) 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: 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(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: 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: 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: 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 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]), (39,248,[0_1|2]), (39,266,[1_1|2]), (39,290,[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]), (63,392,[0_1|2]), (63,410,[1_1|2]), (63,434,[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]), (69,248,[0_1|2]), (69,266,[1_1|2]), (69,290,[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]), (94,320,[0_1|3]), (94,338,[1_1|3]), (94,362,[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]), (136,320,[0_1|3]), (136,338,[1_1|3]), (136,362,[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]), (142,320,[0_1|3]), (142,338,[1_1|3]), (142,362,[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]), (148,320,[0_1|3]), (148,338,[1_1|3]), (148,362,[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]), (154,320,[0_1|3]), (154,338,[1_1|3]), (154,362,[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]), (248,249,[2_1|2]), (249,250,[3_1|2]), (250,251,[4_1|2]), (251,252,[5_1|2]), (252,253,[1_1|2]), (253,254,[1_1|2]), (254,255,[0_1|2]), (255,256,[1_1|2]), (256,257,[2_1|2]), (257,258,[3_1|2]), (258,259,[4_1|2]), (259,260,[5_1|2]), (260,261,[0_1|2]), (261,262,[1_1|2]), (262,263,[2_1|2]), (263,264,[3_1|2]), (264,265,[4_1|2]), (265,178,[5_1|2]), (265,202,[5_1|2]), (266,267,[2_1|2]), (267,268,[3_1|2]), (268,269,[4_1|2]), (269,270,[5_1|2]), (270,271,[1_1|2]), (271,272,[1_1|2]), (272,273,[0_1|2]), (273,274,[1_1|2]), (274,275,[2_1|2]), (275,276,[3_1|2]), (276,277,[4_1|2]), (277,278,[5_1|2]), (278,279,[0_1|2]), (279,280,[1_1|2]), (280,281,[2_1|2]), (281,282,[3_1|2]), (282,283,[4_1|2]), (283,284,[5_1|2]), (284,285,[0_1|2]), (285,286,[1_1|2]), (286,287,[2_1|2]), (287,288,[3_1|2]), (288,289,[4_1|2]), (289,178,[5_1|2]), (289,202,[5_1|2]), (290,291,[2_1|2]), (291,292,[3_1|2]), (292,293,[4_1|2]), (293,294,[5_1|2]), (294,295,[1_1|2]), (295,296,[1_1|2]), (296,297,[0_1|2]), (297,298,[1_1|2]), (298,299,[2_1|2]), (299,300,[3_1|2]), (300,301,[4_1|2]), (301,302,[5_1|2]), (302,303,[0_1|2]), (303,304,[1_1|2]), (304,305,[2_1|2]), (305,306,[3_1|2]), (306,307,[4_1|2]), (307,308,[5_1|2]), (308,309,[0_1|2]), (309,310,[1_1|2]), (310,311,[2_1|2]), (311,312,[3_1|2]), (312,313,[4_1|2]), (313,314,[5_1|2]), (314,315,[0_1|2]), (315,316,[1_1|2]), (316,317,[2_1|2]), (317,318,[3_1|2]), (318,319,[4_1|2]), (319,178,[5_1|2]), (319,202,[5_1|2]), (320,321,[2_1|3]), (321,322,[3_1|3]), (322,323,[4_1|3]), (323,324,[5_1|3]), (324,325,[1_1|3]), (325,326,[1_1|3]), (326,327,[0_1|3]), (326,536,[0_1|4]), (326,554,[1_1|4]), (326,578,[1_1|4]), (327,328,[1_1|3]), (328,329,[2_1|3]), (329,330,[3_1|3]), (330,331,[4_1|3]), (331,332,[5_1|3]), (332,333,[0_1|3]), (332,464,[0_1|4]), (332,482,[1_1|4]), (332,506,[1_1|4]), (333,334,[1_1|3]), (334,335,[2_1|3]), (335,336,[3_1|3]), (336,337,[4_1|3]), (337,106,[5_1|3]), (337,130,[5_1|3]), (337,338,[5_1|3]), (337,362,[5_1|3]), (337,112,[5_1|3]), (337,136,[5_1|3]), (338,339,[2_1|3]), (339,340,[3_1|3]), (340,341,[4_1|3]), (341,342,[5_1|3]), (342,343,[1_1|3]), (343,344,[1_1|3]), (344,345,[0_1|3]), (344,608,[0_1|4]), (344,626,[1_1|4]), (344,650,[1_1|4]), (345,346,[1_1|3]), (346,347,[2_1|3]), (347,348,[3_1|3]), (348,349,[4_1|3]), (349,350,[5_1|3]), (350,351,[0_1|3]), (350,536,[0_1|4]), (350,554,[1_1|4]), (350,578,[1_1|4]), (351,352,[1_1|3]), (352,353,[2_1|3]), (353,354,[3_1|3]), (354,355,[4_1|3]), (355,356,[5_1|3]), (356,357,[0_1|3]), (356,464,[0_1|4]), (356,482,[1_1|4]), (356,506,[1_1|4]), (357,358,[1_1|3]), (358,359,[2_1|3]), (359,360,[3_1|3]), (360,361,[4_1|3]), (361,106,[5_1|3]), (361,130,[5_1|3]), (361,338,[5_1|3]), (361,362,[5_1|3]), (361,112,[5_1|3]), (361,136,[5_1|3]), (362,363,[2_1|3]), (363,364,[3_1|3]), (364,365,[4_1|3]), (365,366,[5_1|3]), (366,367,[1_1|3]), (367,368,[1_1|3]), (368,369,[0_1|3]), (368,680,[0_1|4]), (368,698,[1_1|4]), (368,722,[1_1|4]), (369,370,[1_1|3]), (370,371,[2_1|3]), (371,372,[3_1|3]), (372,373,[4_1|3]), (373,374,[5_1|3]), (374,375,[0_1|3]), (374,608,[0_1|4]), (374,626,[1_1|4]), (374,650,[1_1|4]), (375,376,[1_1|3]), (376,377,[2_1|3]), (377,378,[3_1|3]), (378,379,[4_1|3]), (379,380,[5_1|3]), (380,381,[0_1|3]), (380,536,[0_1|4]), (380,554,[1_1|4]), (380,578,[1_1|4]), (381,382,[1_1|3]), (382,383,[2_1|3]), (383,384,[3_1|3]), (384,385,[4_1|3]), (385,386,[5_1|3]), (386,387,[0_1|3]), (386,464,[0_1|4]), (386,482,[1_1|4]), (386,506,[1_1|4]), (387,388,[1_1|3]), (388,389,[2_1|3]), (389,390,[3_1|3]), (390,391,[4_1|3]), (391,106,[5_1|3]), (391,130,[5_1|3]), (391,338,[5_1|3]), (391,362,[5_1|3]), (391,112,[5_1|3]), (391,136,[5_1|3]), (392,393,[2_1|2]), (393,394,[3_1|2]), (394,395,[4_1|2]), (395,396,[5_1|2]), (396,397,[1_1|2]), (397,398,[1_1|2]), (398,399,[0_1|2]), (399,400,[1_1|2]), (400,401,[2_1|2]), (401,402,[3_1|2]), (402,403,[4_1|2]), (403,404,[5_1|2]), (404,405,[0_1|2]), (405,406,[1_1|2]), (406,407,[2_1|2]), (407,408,[3_1|2]), (408,409,[4_1|2]), (409,266,[5_1|2]), (409,290,[5_1|2]), (410,411,[2_1|2]), (411,412,[3_1|2]), (412,413,[4_1|2]), (413,414,[5_1|2]), (414,415,[1_1|2]), (415,416,[1_1|2]), (416,417,[0_1|2]), (417,418,[1_1|2]), (418,419,[2_1|2]), (419,420,[3_1|2]), (420,421,[4_1|2]), (421,422,[5_1|2]), (422,423,[0_1|2]), (423,424,[1_1|2]), (424,425,[2_1|2]), (425,426,[3_1|2]), (426,427,[4_1|2]), (427,428,[5_1|2]), (428,429,[0_1|2]), (429,430,[1_1|2]), (430,431,[2_1|2]), (431,432,[3_1|2]), (432,433,[4_1|2]), (433,266,[5_1|2]), (433,290,[5_1|2]), (434,435,[2_1|2]), (435,436,[3_1|2]), (436,437,[4_1|2]), (437,438,[5_1|2]), (438,439,[1_1|2]), (439,440,[1_1|2]), (440,441,[0_1|2]), (441,442,[1_1|2]), (442,443,[2_1|2]), (443,444,[3_1|2]), (444,445,[4_1|2]), (445,446,[5_1|2]), (446,447,[0_1|2]), (447,448,[1_1|2]), (448,449,[2_1|2]), (449,450,[3_1|2]), (450,451,[4_1|2]), (451,452,[5_1|2]), (452,453,[0_1|2]), (453,454,[1_1|2]), (454,455,[2_1|2]), (455,456,[3_1|2]), (456,457,[4_1|2]), (457,458,[5_1|2]), (458,459,[0_1|2]), (459,460,[1_1|2]), (460,461,[2_1|2]), (461,462,[3_1|2]), (462,463,[4_1|2]), (463,266,[5_1|2]), (463,290,[5_1|2]), (464,465,[2_1|4]), (465,466,[3_1|4]), (466,467,[4_1|4]), (467,468,[5_1|4]), (468,469,[1_1|4]), (469,470,[1_1|4]), (470,471,[0_1|4]), (471,472,[1_1|4]), (472,473,[2_1|4]), (473,474,[3_1|4]), (474,475,[4_1|4]), (475,476,[5_1|4]), (476,477,[0_1|4]), (477,478,[1_1|4]), (478,479,[2_1|4]), (479,480,[3_1|4]), (480,481,[4_1|4]), (481,338,[5_1|4]), (481,362,[5_1|4]), (482,483,[2_1|4]), (483,484,[3_1|4]), (484,485,[4_1|4]), (485,486,[5_1|4]), (486,487,[1_1|4]), (487,488,[1_1|4]), (488,489,[0_1|4]), (489,490,[1_1|4]), (490,491,[2_1|4]), (491,492,[3_1|4]), (492,493,[4_1|4]), (493,494,[5_1|4]), (494,495,[0_1|4]), (495,496,[1_1|4]), (496,497,[2_1|4]), (497,498,[3_1|4]), (498,499,[4_1|4]), (499,500,[5_1|4]), (500,501,[0_1|4]), (501,502,[1_1|4]), (502,503,[2_1|4]), (503,504,[3_1|4]), (504,505,[4_1|4]), (505,338,[5_1|4]), (505,362,[5_1|4]), (506,507,[2_1|4]), (507,508,[3_1|4]), (508,509,[4_1|4]), (509,510,[5_1|4]), (510,511,[1_1|4]), (511,512,[1_1|4]), (512,513,[0_1|4]), (513,514,[1_1|4]), (514,515,[2_1|4]), (515,516,[3_1|4]), (516,517,[4_1|4]), (517,518,[5_1|4]), (518,519,[0_1|4]), (519,520,[1_1|4]), (520,521,[2_1|4]), (521,522,[3_1|4]), (522,523,[4_1|4]), (523,524,[5_1|4]), (524,525,[0_1|4]), (525,526,[1_1|4]), (526,527,[2_1|4]), (527,528,[3_1|4]), (528,529,[4_1|4]), (529,530,[5_1|4]), (530,531,[0_1|4]), (531,532,[1_1|4]), (532,533,[2_1|4]), (533,534,[3_1|4]), (534,535,[4_1|4]), (535,338,[5_1|4]), (535,362,[5_1|4]), (536,537,[2_1|4]), (537,538,[3_1|4]), (538,539,[4_1|4]), (539,540,[5_1|4]), (540,541,[1_1|4]), (541,542,[1_1|4]), (542,543,[0_1|4]), (543,544,[1_1|4]), (544,545,[2_1|4]), (545,546,[3_1|4]), (546,547,[4_1|4]), (547,548,[5_1|4]), (548,549,[0_1|4]), (549,550,[1_1|4]), (550,551,[2_1|4]), (551,552,[3_1|4]), (552,553,[4_1|4]), (553,482,[5_1|4]), (553,506,[5_1|4]), (554,555,[2_1|4]), (555,556,[3_1|4]), (556,557,[4_1|4]), (557,558,[5_1|4]), (558,559,[1_1|4]), (559,560,[1_1|4]), (560,561,[0_1|4]), (561,562,[1_1|4]), (562,563,[2_1|4]), (563,564,[3_1|4]), (564,565,[4_1|4]), (565,566,[5_1|4]), (566,567,[0_1|4]), (567,568,[1_1|4]), (568,569,[2_1|4]), (569,570,[3_1|4]), (570,571,[4_1|4]), (571,572,[5_1|4]), (572,573,[0_1|4]), (573,574,[1_1|4]), (574,575,[2_1|4]), (575,576,[3_1|4]), (576,577,[4_1|4]), (577,482,[5_1|4]), (577,506,[5_1|4]), (578,579,[2_1|4]), (579,580,[3_1|4]), (580,581,[4_1|4]), (581,582,[5_1|4]), (582,583,[1_1|4]), (583,584,[1_1|4]), (584,585,[0_1|4]), (585,586,[1_1|4]), (586,587,[2_1|4]), (587,588,[3_1|4]), (588,589,[4_1|4]), (589,590,[5_1|4]), (590,591,[0_1|4]), (591,592,[1_1|4]), (592,593,[2_1|4]), (593,594,[3_1|4]), (594,595,[4_1|4]), (595,596,[5_1|4]), (596,597,[0_1|4]), (597,598,[1_1|4]), (598,599,[2_1|4]), (599,600,[3_1|4]), (600,601,[4_1|4]), (601,602,[5_1|4]), (602,603,[0_1|4]), (603,604,[1_1|4]), (604,605,[2_1|4]), (605,606,[3_1|4]), (606,607,[4_1|4]), (607,482,[5_1|4]), (607,506,[5_1|4]), (608,609,[2_1|4]), (609,610,[3_1|4]), (610,611,[4_1|4]), (611,612,[5_1|4]), (612,613,[1_1|4]), (613,614,[1_1|4]), (614,615,[0_1|4]), (615,616,[1_1|4]), (616,617,[2_1|4]), (617,618,[3_1|4]), (618,619,[4_1|4]), (619,620,[5_1|4]), (620,621,[0_1|4]), (621,622,[1_1|4]), (622,623,[2_1|4]), (623,624,[3_1|4]), (624,625,[4_1|4]), (625,554,[5_1|4]), (625,578,[5_1|4]), (626,627,[2_1|4]), (627,628,[3_1|4]), (628,629,[4_1|4]), (629,630,[5_1|4]), (630,631,[1_1|4]), (631,632,[1_1|4]), (632,633,[0_1|4]), (633,634,[1_1|4]), (634,635,[2_1|4]), (635,636,[3_1|4]), (636,637,[4_1|4]), (637,638,[5_1|4]), (638,639,[0_1|4]), (639,640,[1_1|4]), (640,641,[2_1|4]), (641,642,[3_1|4]), (642,643,[4_1|4]), (643,644,[5_1|4]), (644,645,[0_1|4]), (645,646,[1_1|4]), (646,647,[2_1|4]), (647,648,[3_1|4]), (648,649,[4_1|4]), (649,554,[5_1|4]), (649,578,[5_1|4]), (650,651,[2_1|4]), (651,652,[3_1|4]), (652,653,[4_1|4]), (653,654,[5_1|4]), (654,655,[1_1|4]), (655,656,[1_1|4]), (656,657,[0_1|4]), (657,658,[1_1|4]), (658,659,[2_1|4]), (659,660,[3_1|4]), (660,661,[4_1|4]), (661,662,[5_1|4]), (662,663,[0_1|4]), (663,664,[1_1|4]), (664,665,[2_1|4]), (665,666,[3_1|4]), (666,667,[4_1|4]), (667,668,[5_1|4]), (668,669,[0_1|4]), (669,670,[1_1|4]), (670,671,[2_1|4]), (671,672,[3_1|4]), (672,673,[4_1|4]), (673,674,[5_1|4]), (674,675,[0_1|4]), (675,676,[1_1|4]), (676,677,[2_1|4]), (677,678,[3_1|4]), (678,679,[4_1|4]), (679,554,[5_1|4]), (679,578,[5_1|4]), (680,681,[2_1|4]), (681,682,[3_1|4]), (682,683,[4_1|4]), (683,684,[5_1|4]), (684,685,[1_1|4]), (685,686,[1_1|4]), (686,687,[0_1|4]), (687,688,[1_1|4]), (688,689,[2_1|4]), (689,690,[3_1|4]), (690,691,[4_1|4]), (691,692,[5_1|4]), (692,693,[0_1|4]), (693,694,[1_1|4]), (694,695,[2_1|4]), (695,696,[3_1|4]), (696,697,[4_1|4]), (697,626,[5_1|4]), (697,650,[5_1|4]), (698,699,[2_1|4]), (699,700,[3_1|4]), (700,701,[4_1|4]), (701,702,[5_1|4]), (702,703,[1_1|4]), (703,704,[1_1|4]), (704,705,[0_1|4]), (705,706,[1_1|4]), (706,707,[2_1|4]), (707,708,[3_1|4]), (708,709,[4_1|4]), (709,710,[5_1|4]), (710,711,[0_1|4]), (711,712,[1_1|4]), (712,713,[2_1|4]), (713,714,[3_1|4]), (714,715,[4_1|4]), (715,716,[5_1|4]), (716,717,[0_1|4]), (717,718,[1_1|4]), (718,719,[2_1|4]), (719,720,[3_1|4]), (720,721,[4_1|4]), (721,626,[5_1|4]), (721,650,[5_1|4]), (722,723,[2_1|4]), (723,724,[3_1|4]), (724,725,[4_1|4]), (725,726,[5_1|4]), (726,727,[1_1|4]), (727,728,[1_1|4]), (728,729,[0_1|4]), (729,730,[1_1|4]), (730,731,[2_1|4]), (731,732,[3_1|4]), (732,733,[4_1|4]), (733,734,[5_1|4]), (734,735,[0_1|4]), (735,736,[1_1|4]), (736,737,[2_1|4]), (737,738,[3_1|4]), (738,739,[4_1|4]), (739,740,[5_1|4]), (740,741,[0_1|4]), (741,742,[1_1|4]), (742,743,[2_1|4]), (743,744,[3_1|4]), (744,745,[4_1|4]), (745,746,[5_1|4]), (746,747,[0_1|4]), (747,748,[1_1|4]), (748,749,[2_1|4]), (749,750,[3_1|4]), (750,751,[4_1|4]), (751,626,[5_1|4]), (751,650,[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 (innermost) 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: INNERMOST ---------------------------------------- (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 (innermost) 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: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) 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: INNERMOST