/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 84 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 54 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 4 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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(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)) ---------------------------------------- (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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(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(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)) 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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(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(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)) 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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(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)) 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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(263,279,[2_1|3]), (263,79,[2_1|3]), (263,88,[2_1|3]), (263,100,[2_1|3]), (263,115,[2_1|3]), (264,265,[2_1|3]), (265,266,[1_1|3]), (266,267,[1_1|3]), (267,268,[0_1|3]), (267,567,[1_1|4]), (267,576,[1_1|4]), (267,588,[1_1|4]), (267,603,[1_1|4]), (268,269,[1_1|3]), (269,270,[2_1|3]), (270,271,[0_1|3]), (270,513,[1_1|4]), (270,522,[1_1|4]), (270,534,[1_1|4]), (270,549,[1_1|4]), (271,272,[1_1|3]), (272,273,[2_1|3]), (273,274,[0_1|3]), (273,459,[1_1|4]), (273,468,[1_1|4]), (273,480,[1_1|4]), (273,495,[1_1|4]), (274,275,[1_1|3]), (275,276,[2_1|3]), (276,277,[0_1|3]), (276,405,[1_1|4]), (276,414,[1_1|4]), (276,426,[1_1|4]), (276,441,[1_1|4]), (277,278,[1_1|3]), (278,76,[2_1|3]), (278,85,[2_1|3]), (278,97,[2_1|3]), (278,112,[2_1|3]), (278,243,[2_1|3]), (278,252,[2_1|3]), (278,264,[2_1|3]), (278,279,[2_1|3]), (278,79,[2_1|3]), (278,88,[2_1|3]), (278,100,[2_1|3]), (278,115,[2_1|3]), (279,280,[2_1|3]), (280,281,[1_1|3]), (281,282,[1_1|3]), (282,283,[0_1|3]), (282,621,[1_1|4]), (282,630,[1_1|4]), (282,642,[1_1|4]), (282,657,[1_1|4]), (283,284,[1_1|3]), (284,285,[2_1|3]), (285,286,[0_1|3]), (285,567,[1_1|4]), (285,576,[1_1|4]), (285,588,[1_1|4]), (285,603,[1_1|4]), (286,287,[1_1|3]), (287,288,[2_1|3]), (288,289,[0_1|3]), (288,513,[1_1|4]), (288,522,[1_1|4]), (288,534,[1_1|4]), (288,549,[1_1|4]), (289,290,[1_1|3]), (290,291,[2_1|3]), (291,292,[0_1|3]), (291,459,[1_1|4]), (291,468,[1_1|4]), (291,480,[1_1|4]), (291,495,[1_1|4]), (292,293,[1_1|3]), (293,294,[2_1|3]), (294,295,[0_1|3]), (294,405,[1_1|4]), (294,414,[1_1|4]), (294,426,[1_1|4]), (294,441,[1_1|4]), (295,296,[1_1|3]), (296,76,[2_1|3]), (296,85,[2_1|3]), (296,97,[2_1|3]), (296,112,[2_1|3]), (296,243,[2_1|3]), (296,252,[2_1|3]), (296,264,[2_1|3]), (296,279,[2_1|3]), (296,79,[2_1|3]), (296,88,[2_1|3]), (296,100,[2_1|3]), (296,115,[2_1|3]), (297,298,[2_1|2]), (298,299,[1_1|2]), (299,300,[1_1|2]), (300,301,[0_1|2]), (301,302,[1_1|2]), (302,303,[2_1|2]), (303,304,[0_1|2]), (304,305,[1_1|2]), (305,189,[2_1|2]), (305,198,[2_1|2]), (305,210,[2_1|2]), (305,225,[2_1|2]), (306,307,[2_1|2]), (307,308,[1_1|2]), (308,309,[1_1|2]), (309,310,[0_1|2]), (310,311,[1_1|2]), (311,312,[2_1|2]), (312,313,[0_1|2]), (313,314,[1_1|2]), (314,315,[2_1|2]), (315,316,[0_1|2]), (316,317,[1_1|2]), (317,189,[2_1|2]), (317,198,[2_1|2]), (317,210,[2_1|2]), (317,225,[2_1|2]), (318,319,[2_1|2]), (319,320,[1_1|2]), (320,321,[1_1|2]), (321,322,[0_1|2]), (322,323,[1_1|2]), (323,324,[2_1|2]), (324,325,[0_1|2]), (325,326,[1_1|2]), (326,327,[2_1|2]), (327,328,[0_1|2]), (328,329,[1_1|2]), (329,330,[2_1|2]), (330,331,[0_1|2]), (331,332,[1_1|2]), (332,189,[2_1|2]), (332,198,[2_1|2]), (332,210,[2_1|2]), (332,225,[2_1|2]), (333,334,[2_1|2]), (334,335,[1_1|2]), (335,336,[1_1|2]), (336,337,[0_1|2]), (337,338,[1_1|2]), (338,339,[2_1|2]), (339,340,[0_1|2]), (340,341,[1_1|2]), (341,342,[2_1|2]), (342,343,[0_1|2]), (343,344,[1_1|2]), (344,345,[2_1|2]), (345,346,[0_1|2]), (346,347,[1_1|2]), (347,348,[2_1|2]), (348,349,[0_1|2]), (349,350,[1_1|2]), (350,189,[2_1|2]), (350,198,[2_1|2]), (350,210,[2_1|2]), (350,225,[2_1|2]), (351,352,[2_1|2]), (352,353,[1_1|2]), (353,354,[1_1|2]), (354,355,[0_1|2]), (355,356,[1_1|2]), (356,357,[2_1|2]), (357,358,[0_1|2]), (358,359,[1_1|2]), (359,297,[2_1|2]), (359,306,[2_1|2]), (359,318,[2_1|2]), (359,333,[2_1|2]), (360,361,[2_1|2]), (361,362,[1_1|2]), (362,363,[1_1|2]), (363,364,[0_1|2]), (364,365,[1_1|2]), (365,366,[2_1|2]), (366,367,[0_1|2]), (367,368,[1_1|2]), (368,369,[2_1|2]), (369,370,[0_1|2]), (370,371,[1_1|2]), (371,297,[2_1|2]), (371,306,[2_1|2]), (371,318,[2_1|2]), (371,333,[2_1|2]), (372,373,[2_1|2]), (373,374,[1_1|2]), (374,375,[1_1|2]), (375,376,[0_1|2]), (376,377,[1_1|2]), (377,378,[2_1|2]), (378,379,[0_1|2]), (379,380,[1_1|2]), (380,381,[2_1|2]), (381,382,[0_1|2]), (382,383,[1_1|2]), (383,384,[2_1|2]), (384,385,[0_1|2]), (385,386,[1_1|2]), (386,297,[2_1|2]), (386,306,[2_1|2]), (386,318,[2_1|2]), (386,333,[2_1|2]), (387,388,[2_1|2]), (388,389,[1_1|2]), (389,390,[1_1|2]), (390,391,[0_1|2]), (391,392,[1_1|2]), (392,393,[2_1|2]), (393,394,[0_1|2]), (394,395,[1_1|2]), (395,396,[2_1|2]), (396,397,[0_1|2]), (397,398,[1_1|2]), (398,399,[2_1|2]), (399,400,[0_1|2]), (400,401,[1_1|2]), (401,402,[2_1|2]), (402,403,[0_1|2]), (403,404,[1_1|2]), (404,297,[2_1|2]), (404,306,[2_1|2]), (404,318,[2_1|2]), (404,333,[2_1|2]), (405,406,[2_1|4]), (406,407,[1_1|4]), (407,408,[1_1|4]), (408,409,[0_1|4]), (409,410,[1_1|4]), (410,411,[2_1|4]), (411,412,[0_1|4]), (412,413,[1_1|4]), (413,243,[2_1|4]), (413,252,[2_1|4]), (413,264,[2_1|4]), (413,279,[2_1|4]), (414,415,[2_1|4]), (415,416,[1_1|4]), (416,417,[1_1|4]), (417,418,[0_1|4]), (418,419,[1_1|4]), (419,420,[2_1|4]), (420,421,[0_1|4]), (421,422,[1_1|4]), (422,423,[2_1|4]), (423,424,[0_1|4]), (424,425,[1_1|4]), (425,243,[2_1|4]), (425,252,[2_1|4]), (425,264,[2_1|4]), (425,279,[2_1|4]), (426,427,[2_1|4]), (427,428,[1_1|4]), (428,429,[1_1|4]), (429,430,[0_1|4]), (430,431,[1_1|4]), (431,432,[2_1|4]), (432,433,[0_1|4]), (433,434,[1_1|4]), (434,435,[2_1|4]), (435,436,[0_1|4]), (436,437,[1_1|4]), (437,438,[2_1|4]), (438,439,[0_1|4]), (439,440,[1_1|4]), (440,243,[2_1|4]), (440,252,[2_1|4]), (440,264,[2_1|4]), (440,279,[2_1|4]), (441,442,[2_1|4]), (442,443,[1_1|4]), (443,444,[1_1|4]), (444,445,[0_1|4]), (445,446,[1_1|4]), (446,447,[2_1|4]), (447,448,[0_1|4]), (448,449,[1_1|4]), (449,450,[2_1|4]), (450,451,[0_1|4]), (451,452,[1_1|4]), (452,453,[2_1|4]), (453,454,[0_1|4]), (454,455,[1_1|4]), (455,456,[2_1|4]), (456,457,[0_1|4]), (457,458,[1_1|4]), (458,243,[2_1|4]), (458,252,[2_1|4]), (458,264,[2_1|4]), (458,279,[2_1|4]), (459,460,[2_1|4]), (460,461,[1_1|4]), (461,462,[1_1|4]), (462,463,[0_1|4]), (463,464,[1_1|4]), (464,465,[2_1|4]), (465,466,[0_1|4]), (466,467,[1_1|4]), (467,405,[2_1|4]), (467,414,[2_1|4]), (467,426,[2_1|4]), (467,441,[2_1|4]), (468,469,[2_1|4]), (469,470,[1_1|4]), (470,471,[1_1|4]), (471,472,[0_1|4]), (472,473,[1_1|4]), (473,474,[2_1|4]), (474,475,[0_1|4]), (475,476,[1_1|4]), (476,477,[2_1|4]), (477,478,[0_1|4]), (478,479,[1_1|4]), (479,405,[2_1|4]), (479,414,[2_1|4]), (479,426,[2_1|4]), (479,441,[2_1|4]), (480,481,[2_1|4]), (481,482,[1_1|4]), (482,483,[1_1|4]), (483,484,[0_1|4]), (484,485,[1_1|4]), (485,486,[2_1|4]), (486,487,[0_1|4]), (487,488,[1_1|4]), (488,489,[2_1|4]), (489,490,[0_1|4]), (490,491,[1_1|4]), (491,492,[2_1|4]), (492,493,[0_1|4]), (493,494,[1_1|4]), (494,405,[2_1|4]), (494,414,[2_1|4]), (494,426,[2_1|4]), (494,441,[2_1|4]), (495,496,[2_1|4]), (496,497,[1_1|4]), (497,498,[1_1|4]), (498,499,[0_1|4]), (499,500,[1_1|4]), (500,501,[2_1|4]), (501,502,[0_1|4]), (502,503,[1_1|4]), (503,504,[2_1|4]), (504,505,[0_1|4]), (505,506,[1_1|4]), (506,507,[2_1|4]), (507,508,[0_1|4]), (508,509,[1_1|4]), (509,510,[2_1|4]), (510,511,[0_1|4]), (511,512,[1_1|4]), (512,405,[2_1|4]), (512,414,[2_1|4]), (512,426,[2_1|4]), (512,441,[2_1|4]), (513,514,[2_1|4]), (514,515,[1_1|4]), (515,516,[1_1|4]), (516,517,[0_1|4]), (517,518,[1_1|4]), (518,519,[2_1|4]), (519,520,[0_1|4]), (520,521,[1_1|4]), (521,459,[2_1|4]), (521,468,[2_1|4]), (521,480,[2_1|4]), (521,495,[2_1|4]), (522,523,[2_1|4]), (523,524,[1_1|4]), (524,525,[1_1|4]), (525,526,[0_1|4]), (526,527,[1_1|4]), (527,528,[2_1|4]), (528,529,[0_1|4]), (529,530,[1_1|4]), (530,531,[2_1|4]), (531,532,[0_1|4]), (532,533,[1_1|4]), (533,459,[2_1|4]), (533,468,[2_1|4]), (533,480,[2_1|4]), (533,495,[2_1|4]), (534,535,[2_1|4]), (535,536,[1_1|4]), (536,537,[1_1|4]), (537,538,[0_1|4]), (538,539,[1_1|4]), (539,540,[2_1|4]), (540,541,[0_1|4]), (541,542,[1_1|4]), (542,543,[2_1|4]), (543,544,[0_1|4]), (544,545,[1_1|4]), (545,546,[2_1|4]), (546,547,[0_1|4]), (547,548,[1_1|4]), (548,459,[2_1|4]), (548,468,[2_1|4]), (548,480,[2_1|4]), (548,495,[2_1|4]), (549,550,[2_1|4]), (550,551,[1_1|4]), (551,552,[1_1|4]), (552,553,[0_1|4]), (553,554,[1_1|4]), (554,555,[2_1|4]), (555,556,[0_1|4]), (556,557,[1_1|4]), (557,558,[2_1|4]), (558,559,[0_1|4]), (559,560,[1_1|4]), (560,561,[2_1|4]), (561,562,[0_1|4]), (562,563,[1_1|4]), (563,564,[2_1|4]), (564,565,[0_1|4]), (565,566,[1_1|4]), (566,459,[2_1|4]), (566,468,[2_1|4]), (566,480,[2_1|4]), (566,495,[2_1|4]), (567,568,[2_1|4]), (568,569,[1_1|4]), (569,570,[1_1|4]), (570,571,[0_1|4]), (571,572,[1_1|4]), (572,573,[2_1|4]), (573,574,[0_1|4]), (574,575,[1_1|4]), (575,513,[2_1|4]), (575,522,[2_1|4]), (575,534,[2_1|4]), (575,549,[2_1|4]), (576,577,[2_1|4]), (577,578,[1_1|4]), (578,579,[1_1|4]), (579,580,[0_1|4]), (580,581,[1_1|4]), (581,582,[2_1|4]), (582,583,[0_1|4]), (583,584,[1_1|4]), (584,585,[2_1|4]), (585,586,[0_1|4]), (586,587,[1_1|4]), (587,513,[2_1|4]), (587,522,[2_1|4]), (587,534,[2_1|4]), (587,549,[2_1|4]), (588,589,[2_1|4]), (589,590,[1_1|4]), (590,591,[1_1|4]), (591,592,[0_1|4]), (592,593,[1_1|4]), (593,594,[2_1|4]), (594,595,[0_1|4]), (595,596,[1_1|4]), (596,597,[2_1|4]), (597,598,[0_1|4]), (598,599,[1_1|4]), (599,600,[2_1|4]), (600,601,[0_1|4]), (601,602,[1_1|4]), (602,513,[2_1|4]), (602,522,[2_1|4]), (602,534,[2_1|4]), (602,549,[2_1|4]), (603,604,[2_1|4]), (604,605,[1_1|4]), (605,606,[1_1|4]), (606,607,[0_1|4]), (607,608,[1_1|4]), (608,609,[2_1|4]), (609,610,[0_1|4]), (610,611,[1_1|4]), (611,612,[2_1|4]), (612,613,[0_1|4]), (613,614,[1_1|4]), (614,615,[2_1|4]), (615,616,[0_1|4]), (616,617,[1_1|4]), (617,618,[2_1|4]), (618,619,[0_1|4]), (619,620,[1_1|4]), (620,513,[2_1|4]), (620,522,[2_1|4]), (620,534,[2_1|4]), (620,549,[2_1|4]), (621,622,[2_1|4]), (622,623,[1_1|4]), (623,624,[1_1|4]), (624,625,[0_1|4]), (625,626,[1_1|4]), (626,627,[2_1|4]), (627,628,[0_1|4]), (628,629,[1_1|4]), (629,567,[2_1|4]), (629,576,[2_1|4]), (629,588,[2_1|4]), (629,603,[2_1|4]), (630,631,[2_1|4]), (631,632,[1_1|4]), (632,633,[1_1|4]), (633,634,[0_1|4]), (634,635,[1_1|4]), (635,636,[2_1|4]), (636,637,[0_1|4]), (637,638,[1_1|4]), (638,639,[2_1|4]), (639,640,[0_1|4]), (640,641,[1_1|4]), (641,567,[2_1|4]), (641,576,[2_1|4]), (641,588,[2_1|4]), (641,603,[2_1|4]), (642,643,[2_1|4]), (643,644,[1_1|4]), (644,645,[1_1|4]), (645,646,[0_1|4]), (646,647,[1_1|4]), (647,648,[2_1|4]), (648,649,[0_1|4]), (649,650,[1_1|4]), (650,651,[2_1|4]), (651,652,[0_1|4]), (652,653,[1_1|4]), (653,654,[2_1|4]), (654,655,[0_1|4]), (655,656,[1_1|4]), (656,567,[2_1|4]), (656,576,[2_1|4]), (656,588,[2_1|4]), (656,603,[2_1|4]), (657,658,[2_1|4]), (658,659,[1_1|4]), (659,660,[1_1|4]), (660,661,[0_1|4]), (661,662,[1_1|4]), (662,663,[2_1|4]), (663,664,[0_1|4]), (664,665,[1_1|4]), (665,666,[2_1|4]), (666,667,[0_1|4]), (667,668,[1_1|4]), (668,669,[2_1|4]), (669,670,[0_1|4]), (670,671,[1_1|4]), (671,672,[2_1|4]), (672,673,[0_1|4]), (673,674,[1_1|4]), (674,567,[2_1|4]), (674,576,[2_1|4]), (674,588,[2_1|4]), (674,603,[2_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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(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(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)) 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(1(x1)))) ->^+ 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(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]. 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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(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(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)) 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(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(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(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)) Rewrite Strategy: INNERMOST