WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 77 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 0 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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(17,198,[1_1|2]), (18,19,[1_1|1]), (19,20,[2_1|1]), (20,21,[3_1|1]), (21,22,[4_1|1]), (22,23,[5_1|1]), (23,24,[0_1|1]), (23,11,[0_1|1]), (23,29,[1_1|1]), (23,53,[1_1|1]), (24,25,[1_1|1]), (25,26,[2_1|1]), (26,27,[3_1|1]), (27,28,[4_1|1]), (28,10,[5_1|1]), (29,30,[2_1|1]), (30,31,[3_1|1]), (31,32,[4_1|1]), (32,33,[5_1|1]), (33,34,[1_1|1]), (34,35,[1_1|1]), (35,36,[0_1|1]), (35,228,[0_1|2]), (35,246,[1_1|2]), (35,270,[1_1|2]), (36,37,[1_1|1]), (37,38,[2_1|1]), (38,39,[3_1|1]), (39,40,[4_1|1]), (40,41,[5_1|1]), (41,42,[0_1|1]), (41,156,[0_1|2]), (41,174,[1_1|2]), (41,198,[1_1|2]), (42,43,[1_1|1]), (43,44,[2_1|1]), (44,45,[3_1|1]), (45,46,[4_1|1]), (46,47,[5_1|1]), (47,48,[0_1|1]), (47,11,[0_1|1]), (47,29,[1_1|1]), (47,53,[1_1|1]), (48,49,[1_1|1]), (49,50,[2_1|1]), (50,51,[3_1|1]), (51,52,[4_1|1]), (52,10,[5_1|1]), (53,54,[2_1|1]), (54,55,[3_1|1]), (55,56,[4_1|1]), (56,57,[5_1|1]), (57,58,[1_1|1]), (58,59,[1_1|1]), (59,60,[0_1|1]), (59,380,[0_1|2]), (59,398,[1_1|2]), (59,422,[1_1|2]), 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(159,160,[5_1|2]), (160,161,[1_1|2]), (161,162,[1_1|2]), (162,163,[0_1|2]), (163,164,[1_1|2]), (164,165,[2_1|2]), (165,166,[3_1|2]), (166,167,[4_1|2]), (167,168,[5_1|2]), (168,169,[0_1|2]), (169,170,[1_1|2]), (170,171,[2_1|2]), (171,172,[3_1|2]), (172,173,[4_1|2]), (173,29,[5_1|2]), (173,53,[5_1|2]), (174,175,[2_1|2]), (175,176,[3_1|2]), (176,177,[4_1|2]), (177,178,[5_1|2]), (178,179,[1_1|2]), (179,180,[1_1|2]), (180,181,[0_1|2]), (181,182,[1_1|2]), (182,183,[2_1|2]), (183,184,[3_1|2]), (184,185,[4_1|2]), (185,186,[5_1|2]), (186,187,[0_1|2]), (187,188,[1_1|2]), (188,189,[2_1|2]), (189,190,[3_1|2]), (190,191,[4_1|2]), (191,192,[5_1|2]), (192,193,[0_1|2]), (193,194,[1_1|2]), (194,195,[2_1|2]), (195,196,[3_1|2]), (196,197,[4_1|2]), (197,29,[5_1|2]), (197,53,[5_1|2]), (198,199,[2_1|2]), (199,200,[3_1|2]), (200,201,[4_1|2]), (201,202,[5_1|2]), (202,203,[1_1|2]), (203,204,[1_1|2]), (204,205,[0_1|2]), (205,206,[1_1|2]), (206,207,[2_1|2]), (207,208,[3_1|2]), (208,209,[4_1|2]), (209,210,[5_1|2]), (210,211,[0_1|2]), (211,212,[1_1|2]), (212,213,[2_1|2]), (213,214,[3_1|2]), (214,215,[4_1|2]), (215,216,[5_1|2]), (216,217,[0_1|2]), (217,218,[1_1|2]), (218,219,[2_1|2]), (219,220,[3_1|2]), (220,221,[4_1|2]), (221,222,[5_1|2]), (222,223,[0_1|2]), (223,224,[1_1|2]), (224,225,[2_1|2]), (225,226,[3_1|2]), (226,227,[4_1|2]), (227,29,[5_1|2]), (227,53,[5_1|2]), (228,229,[2_1|2]), (229,230,[3_1|2]), (230,231,[4_1|2]), (231,232,[5_1|2]), (232,233,[1_1|2]), (233,234,[1_1|2]), (234,235,[0_1|2]), (235,236,[1_1|2]), (236,237,[2_1|2]), (237,238,[3_1|2]), (238,239,[4_1|2]), (239,240,[5_1|2]), (240,241,[0_1|2]), (241,242,[1_1|2]), (242,243,[2_1|2]), (243,244,[3_1|2]), (244,245,[4_1|2]), (245,174,[5_1|2]), (245,198,[5_1|2]), (246,247,[2_1|2]), (247,248,[3_1|2]), (248,249,[4_1|2]), (249,250,[5_1|2]), (250,251,[1_1|2]), (251,252,[1_1|2]), (252,253,[0_1|2]), (253,254,[1_1|2]), (254,255,[2_1|2]), (255,256,[3_1|2]), (256,257,[4_1|2]), (257,258,[5_1|2]), (258,259,[0_1|2]), (259,260,[1_1|2]), (260,261,[2_1|2]), (261,262,[3_1|2]), (262,263,[4_1|2]), (263,264,[5_1|2]), (264,265,[0_1|2]), (265,266,[1_1|2]), (266,267,[2_1|2]), (267,268,[3_1|2]), (268,269,[4_1|2]), (269,174,[5_1|2]), (269,198,[5_1|2]), (270,271,[2_1|2]), (271,272,[3_1|2]), (272,273,[4_1|2]), (273,274,[5_1|2]), (274,275,[1_1|2]), (275,276,[1_1|2]), (276,277,[0_1|2]), (277,278,[1_1|2]), (278,279,[2_1|2]), (279,280,[3_1|2]), (280,281,[4_1|2]), (281,282,[5_1|2]), (282,283,[0_1|2]), (283,284,[1_1|2]), (284,285,[2_1|2]), (285,286,[3_1|2]), (286,287,[4_1|2]), (287,288,[5_1|2]), (288,289,[0_1|2]), (289,290,[1_1|2]), (290,291,[2_1|2]), (291,292,[3_1|2]), (292,293,[4_1|2]), (293,294,[5_1|2]), (294,295,[0_1|2]), (295,296,[1_1|2]), (296,297,[2_1|2]), (297,298,[3_1|2]), (298,299,[4_1|2]), (299,174,[5_1|2]), (299,198,[5_1|2]), (301,302,[2_1|3]), (302,303,[3_1|3]), (303,304,[4_1|3]), (304,305,[5_1|3]), (305,306,[1_1|3]), (306,307,[1_1|3]), (307,308,[0_1|3]), (307,532,[0_1|4]), (307,550,[1_1|4]), (307,574,[1_1|4]), (308,309,[1_1|3]), (309,310,[2_1|3]), (310,311,[3_1|3]), (311,312,[4_1|3]), (312,313,[5_1|3]), (313,314,[0_1|3]), (313,460,[0_1|4]), (313,478,[1_1|4]), (313,502,[1_1|4]), (314,315,[1_1|3]), (315,316,[2_1|3]), (316,317,[3_1|3]), (317,318,[4_1|3]), (318,102,[5_1|3]), (318,126,[5_1|3]), (318,319,[5_1|3]), (318,343,[5_1|3]), (318,108,[5_1|3]), (318,132,[5_1|3]), (319,320,[2_1|3]), (320,321,[3_1|3]), (321,322,[4_1|3]), (322,323,[5_1|3]), (323,324,[1_1|3]), (324,325,[1_1|3]), (325,326,[0_1|3]), (325,604,[0_1|4]), (325,622,[1_1|4]), (325,646,[1_1|4]), (326,327,[1_1|3]), (327,328,[2_1|3]), (328,329,[3_1|3]), (329,330,[4_1|3]), (330,331,[5_1|3]), (331,332,[0_1|3]), (331,532,[0_1|4]), (331,550,[1_1|4]), (331,574,[1_1|4]), (332,333,[1_1|3]), (333,334,[2_1|3]), (334,335,[3_1|3]), (335,336,[4_1|3]), (336,337,[5_1|3]), (337,338,[0_1|3]), (337,460,[0_1|4]), (337,478,[1_1|4]), (337,502,[1_1|4]), (338,339,[1_1|3]), (339,340,[2_1|3]), (340,341,[3_1|3]), (341,342,[4_1|3]), (342,102,[5_1|3]), (342,126,[5_1|3]), (342,319,[5_1|3]), (342,343,[5_1|3]), (342,108,[5_1|3]), (342,132,[5_1|3]), (343,344,[2_1|3]), (344,345,[3_1|3]), (345,346,[4_1|3]), (346,347,[5_1|3]), (347,348,[1_1|3]), (348,349,[1_1|3]), (349,350,[0_1|3]), (349,676,[0_1|4]), (349,694,[1_1|4]), (349,718,[1_1|4]), (350,351,[1_1|3]), (351,352,[2_1|3]), (352,353,[3_1|3]), (353,354,[4_1|3]), (354,355,[5_1|3]), (355,356,[0_1|3]), (355,604,[0_1|4]), (355,622,[1_1|4]), (355,646,[1_1|4]), (356,357,[1_1|3]), (357,358,[2_1|3]), (358,359,[3_1|3]), (359,360,[4_1|3]), (360,361,[5_1|3]), (361,362,[0_1|3]), (361,532,[0_1|4]), (361,550,[1_1|4]), (361,574,[1_1|4]), (362,363,[1_1|3]), (363,364,[2_1|3]), (364,365,[3_1|3]), (365,366,[4_1|3]), (366,367,[5_1|3]), (367,368,[0_1|3]), (367,460,[0_1|4]), (367,478,[1_1|4]), (367,502,[1_1|4]), (368,369,[1_1|3]), (369,370,[2_1|3]), (370,371,[3_1|3]), (371,372,[4_1|3]), (372,102,[5_1|3]), (372,126,[5_1|3]), (372,319,[5_1|3]), (372,343,[5_1|3]), (372,108,[5_1|3]), (372,132,[5_1|3]), (380,381,[2_1|2]), (381,382,[3_1|2]), (382,383,[4_1|2]), (383,384,[5_1|2]), (384,385,[1_1|2]), (385,386,[1_1|2]), (386,387,[0_1|2]), (387,388,[1_1|2]), (388,389,[2_1|2]), (389,390,[3_1|2]), (390,391,[4_1|2]), (391,392,[5_1|2]), (392,393,[0_1|2]), (393,394,[1_1|2]), (394,395,[2_1|2]), (395,396,[3_1|2]), (396,397,[4_1|2]), (397,246,[5_1|2]), (397,270,[5_1|2]), (398,399,[2_1|2]), (399,400,[3_1|2]), (400,401,[4_1|2]), (401,402,[5_1|2]), (402,403,[1_1|2]), (403,404,[1_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]), (409,410,[5_1|2]), (410,411,[0_1|2]), (411,412,[1_1|2]), (412,413,[2_1|2]), (413,414,[3_1|2]), (414,415,[4_1|2]), (415,416,[5_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]), (421,246,[5_1|2]), (421,270,[5_1|2]), (422,423,[2_1|2]), (423,424,[3_1|2]), (424,425,[4_1|2]), (425,426,[5_1|2]), (426,427,[1_1|2]), (427,428,[1_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]), (433,434,[5_1|2]), (434,435,[0_1|2]), (435,436,[1_1|2]), (436,437,[2_1|2]), (437,438,[3_1|2]), (438,439,[4_1|2]), (439,440,[5_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]), (451,246,[5_1|2]), (451,270,[5_1|2]), (460,461,[2_1|4]), (461,462,[3_1|4]), (462,463,[4_1|4]), (463,464,[5_1|4]), (464,465,[1_1|4]), (465,466,[1_1|4]), (466,467,[0_1|4]), (467,468,[1_1|4]), (468,469,[2_1|4]), (469,470,[3_1|4]), (470,471,[4_1|4]), (471,472,[5_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]), (477,319,[5_1|4]), (477,343,[5_1|4]), (478,479,[2_1|4]), (479,480,[3_1|4]), (480,481,[4_1|4]), (481,482,[5_1|4]), (482,483,[1_1|4]), (483,484,[1_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]), (489,490,[5_1|4]), (490,491,[0_1|4]), (491,492,[1_1|4]), (492,493,[2_1|4]), (493,494,[3_1|4]), (494,495,[4_1|4]), (495,496,[5_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]), (501,319,[5_1|4]), (501,343,[5_1|4]), (502,503,[2_1|4]), (503,504,[3_1|4]), (504,505,[4_1|4]), (505,506,[5_1|4]), (506,507,[1_1|4]), (507,508,[1_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]), (519,520,[5_1|4]), (520,521,[0_1|4]), (521,522,[1_1|4]), (522,523,[2_1|4]), (523,524,[3_1|4]), (524,525,[4_1|4]), (525,526,[5_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]), (531,319,[5_1|4]), (531,343,[5_1|4]), (532,533,[2_1|4]), (533,534,[3_1|4]), (534,535,[4_1|4]), (535,536,[5_1|4]), (536,537,[1_1|4]), (537,538,[1_1|4]), (538,539,[0_1|4]), (539,540,[1_1|4]), (540,541,[2_1|4]), (541,542,[3_1|4]), (542,543,[4_1|4]), (543,544,[5_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]), (549,478,[5_1|4]), (549,502,[5_1|4]), (550,551,[2_1|4]), (551,552,[3_1|4]), (552,553,[4_1|4]), (553,554,[5_1|4]), (554,555,[1_1|4]), (555,556,[1_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]), (561,562,[5_1|4]), (562,563,[0_1|4]), (563,564,[1_1|4]), (564,565,[2_1|4]), (565,566,[3_1|4]), (566,567,[4_1|4]), (567,568,[5_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]), (573,478,[5_1|4]), (573,502,[5_1|4]), (574,575,[2_1|4]), (575,576,[3_1|4]), (576,577,[4_1|4]), (577,578,[5_1|4]), (578,579,[1_1|4]), (579,580,[1_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]), (591,592,[5_1|4]), (592,593,[0_1|4]), (593,594,[1_1|4]), (594,595,[2_1|4]), (595,596,[3_1|4]), (596,597,[4_1|4]), (597,598,[5_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]), (603,478,[5_1|4]), (603,502,[5_1|4]), (604,605,[2_1|4]), (605,606,[3_1|4]), (606,607,[4_1|4]), (607,608,[5_1|4]), (608,609,[1_1|4]), (609,610,[1_1|4]), (610,611,[0_1|4]), (611,612,[1_1|4]), (612,613,[2_1|4]), (613,614,[3_1|4]), (614,615,[4_1|4]), (615,616,[5_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]), (621,550,[5_1|4]), (621,574,[5_1|4]), (622,623,[2_1|4]), (623,624,[3_1|4]), (624,625,[4_1|4]), (625,626,[5_1|4]), (626,627,[1_1|4]), (627,628,[1_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]), (633,634,[5_1|4]), (634,635,[0_1|4]), (635,636,[1_1|4]), (636,637,[2_1|4]), (637,638,[3_1|4]), (638,639,[4_1|4]), (639,640,[5_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]), (645,550,[5_1|4]), (645,574,[5_1|4]), (646,647,[2_1|4]), (647,648,[3_1|4]), (648,649,[4_1|4]), (649,650,[5_1|4]), (650,651,[1_1|4]), (651,652,[1_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]), (663,664,[5_1|4]), (664,665,[0_1|4]), (665,666,[1_1|4]), (666,667,[2_1|4]), (667,668,[3_1|4]), (668,669,[4_1|4]), (669,670,[5_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]), (675,550,[5_1|4]), (675,574,[5_1|4]), (676,677,[2_1|4]), (677,678,[3_1|4]), (678,679,[4_1|4]), (679,680,[5_1|4]), (680,681,[1_1|4]), (681,682,[1_1|4]), (682,683,[0_1|4]), (683,684,[1_1|4]), (684,685,[2_1|4]), (685,686,[3_1|4]), (686,687,[4_1|4]), (687,688,[5_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]), (693,622,[5_1|4]), (693,646,[5_1|4]), (694,695,[2_1|4]), (695,696,[3_1|4]), (696,697,[4_1|4]), (697,698,[5_1|4]), (698,699,[1_1|4]), (699,700,[1_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]), (705,706,[5_1|4]), (706,707,[0_1|4]), (707,708,[1_1|4]), (708,709,[2_1|4]), (709,710,[3_1|4]), (710,711,[4_1|4]), (711,712,[5_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]), (717,622,[5_1|4]), (717,646,[5_1|4]), (718,719,[2_1|4]), (719,720,[3_1|4]), (720,721,[4_1|4]), (721,722,[5_1|4]), (722,723,[1_1|4]), (723,724,[1_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]), (735,736,[5_1|4]), (736,737,[0_1|4]), (737,738,[1_1|4]), (738,739,[2_1|4]), (739,740,[3_1|4]), (740,741,[4_1|4]), (741,742,[5_1|4]), (742,743,[0_1|4]), (743,744,[1_1|4]), (744,745,[2_1|4]), (745,746,[3_1|4]), (746,747,[4_1|4]), (747,622,[5_1|4]), (747,646,[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