KILLED proof of input_7cQP8kKlYM.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 177 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 6 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 250 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 120 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 348 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 125 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 271 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 170 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 338 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 348 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 382 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 419 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 7468 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 2977 ms] (62) CpxRNTS (63) CompletionProof [UPPER BOUND(ID), 0 ms] (64) CpxTypedWeightedCompleteTrs (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (94) CdtProblem (95) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 52 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms] (124) CdtProblem (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 92 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 84 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (142) CdtProblem (143) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 33 ms] (154) CdtProblem (155) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (156) CdtProblem (157) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 135 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (212) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: sort(l) -> st(0', l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0' max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) st(n, l) -> cond1(member(n, l), n, l) cond1(true, n, l) -> cons(n, st(s(n), l)) cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) cond2(true, n, l) -> nil cond2(false, n, l) -> st(s(n), l) member(n, nil) -> false member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) or(x, true) -> true or(x, false) -> x equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) max(nil) -> 0 max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) if(true, u, v) -> u if(false, u, v) -> v S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: sort(l) -> st(0, l) [1] st(n, l) -> cond1(member(n, l), n, l) [1] cond1(true, n, l) -> cons(n, st(s(n), l)) [1] cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) [1] cond2(true, n, l) -> nil [1] cond2(false, n, l) -> st(s(n), l) [1] member(n, nil) -> false [1] member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) [1] or(x, true) -> true [1] or(x, false) -> x [1] equal(0, 0) -> true [1] equal(s(x), 0) -> false [1] equal(0, s(y)) -> false [1] equal(s(x), s(y)) -> equal(x, y) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] max(nil) -> 0 [1] max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) [1] if(true, u, v) -> u [1] if(false, u, v) -> v [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sort(l) -> st(0, l) [1] st(n, l) -> cond1(member(n, l), n, l) [1] cond1(true, n, l) -> cons(n, st(s(n), l)) [1] cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) [1] cond2(true, n, l) -> nil [1] cond2(false, n, l) -> st(s(n), l) [1] member(n, nil) -> false [1] member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) [1] or(x, true) -> true [1] or(x, false) -> x [1] equal(0, 0) -> true [1] equal(s(x), 0) -> false [1] equal(0, s(y)) -> false [1] equal(s(x), s(y)) -> equal(x, y) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] max(nil) -> 0 [1] max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) [1] if(true, u, v) -> u [1] if(false, u, v) -> v [1] The TRS has the following type information: sort :: cons:nil -> cons:nil st :: 0:s -> cons:nil -> cons:nil 0 :: 0:s cond1 :: true:false -> 0:s -> cons:nil -> cons:nil member :: 0:s -> cons:nil -> true:false true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false cond2 :: true:false -> 0:s -> cons:nil -> cons:nil gt :: 0:s -> 0:s -> true:false max :: cons:nil -> 0:s nil :: cons:nil or :: true:false -> true:false -> true:false equal :: 0:s -> 0:s -> true:false if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: sort_1 st_2 cond1_3 cond2_3 (c) The following functions are completely defined: gt_2 max_1 member_2 equal_2 if_3 or_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sort(l) -> st(0, l) [1] st(n, l) -> cond1(member(n, l), n, l) [1] cond1(true, n, l) -> cons(n, st(s(n), l)) [1] cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) [1] cond2(true, n, l) -> nil [1] cond2(false, n, l) -> st(s(n), l) [1] member(n, nil) -> false [1] member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) [1] or(x, true) -> true [1] or(x, false) -> x [1] equal(0, 0) -> true [1] equal(s(x), 0) -> false [1] equal(0, s(y)) -> false [1] equal(s(x), s(y)) -> equal(x, y) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] max(nil) -> 0 [1] max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) [1] if(true, u, v) -> u [1] if(false, u, v) -> v [1] The TRS has the following type information: sort :: cons:nil -> cons:nil st :: 0:s -> cons:nil -> cons:nil 0 :: 0:s cond1 :: true:false -> 0:s -> cons:nil -> cons:nil member :: 0:s -> cons:nil -> true:false true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false cond2 :: true:false -> 0:s -> cons:nil -> cons:nil gt :: 0:s -> 0:s -> true:false max :: cons:nil -> 0:s nil :: cons:nil or :: true:false -> true:false -> true:false equal :: 0:s -> 0:s -> true:false if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sort(l) -> st(0, l) [1] st(n, nil) -> cond1(false, n, nil) [2] st(n, cons(m', l')) -> cond1(or(equal(n, m'), member(n, l')), n, cons(m', l')) [2] cond1(true, n, l) -> cons(n, st(s(n), l)) [1] cond1(false, n, nil) -> cond2(gt(n, 0), n, nil) [2] cond1(false, n, cons(u', l'')) -> cond2(gt(n, if(gt(u', max(l'')), u', max(l''))), n, cons(u', l'')) [2] cond2(true, n, l) -> nil [1] cond2(false, n, l) -> st(s(n), l) [1] member(n, nil) -> false [1] member(0, cons(0, nil)) -> or(true, false) [3] member(0, cons(0, cons(m'', l1))) -> or(true, or(equal(0, m''), member(0, l1))) [3] member(s(x'), cons(0, nil)) -> or(false, false) [3] member(s(x'), cons(0, cons(m1, l2))) -> or(false, or(equal(s(x'), m1), member(s(x'), l2))) [3] member(0, cons(s(y'), nil)) -> or(false, false) [3] member(0, cons(s(y'), cons(m2, l3))) -> or(false, or(equal(0, m2), member(0, l3))) [3] member(s(x''), cons(s(y''), nil)) -> or(equal(x'', y''), false) [3] member(s(x''), cons(s(y''), cons(m3, l4))) -> or(equal(x'', y''), or(equal(s(x''), m3), member(s(x''), l4))) [3] or(x, true) -> true [1] or(x, false) -> x [1] equal(0, 0) -> true [1] equal(s(x), 0) -> false [1] equal(0, s(y)) -> false [1] equal(s(x), s(y)) -> equal(x, y) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] max(nil) -> 0 [1] max(cons(u, nil)) -> if(gt(u, 0), u, 0) [3] max(cons(u, cons(u'', l5))) -> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) [3] if(true, u, v) -> u [1] if(false, u, v) -> v [1] The TRS has the following type information: sort :: cons:nil -> cons:nil st :: 0:s -> cons:nil -> cons:nil 0 :: 0:s cond1 :: true:false -> 0:s -> cons:nil -> cons:nil member :: 0:s -> cons:nil -> true:false true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false cond2 :: true:false -> 0:s -> cons:nil -> cons:nil gt :: 0:s -> 0:s -> true:false max :: cons:nil -> 0:s nil :: cons:nil or :: true:false -> true:false -> true:false equal :: 0:s -> 0:s -> true:false if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(n, if(gt(u', max(l'')), u', max(l''))), n, 1 + u' + l'') :|: n >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z' = n, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(n, 0), n, 0) :|: z'' = 0, n >= 0, z' = n, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + n + st(1 + n, l) :|: n >= 0, z = 1, z' = n, l >= 0, z'' = l cond2(z, z', z'') -{ 1 }-> st(1 + n, l) :|: n >= 0, z' = n, l >= 0, z = 0, z'' = l cond2(z, z', z'') -{ 1 }-> 0 :|: n >= 0, z = 1, z' = n, l >= 0, z'' = l equal(z, z') -{ 1 }-> equal(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0 gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0 if(z, z', z'') -{ 1 }-> u :|: z'' = v, v >= 0, z = 1, z' = u, u >= 0 if(z, z', z'') -{ 1 }-> v :|: z'' = v, v >= 0, z' = u, z = 0, u >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(u, 0), u, 0) :|: z = 1 + u + 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 3 }-> or(equal(x'', y''), or(equal(1 + x'', m3), member(1 + x'', l4))) :|: z = 1 + x'', y'' >= 0, x'' >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(x'', y''), 0) :|: z = 1 + x'', z' = 1 + (1 + y'') + 0, y'' >= 0, x'' >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(1, 0) :|: z' = 1 + 0 + 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + x', m1), member(1 + x', l2))) :|: z = 1 + x', x' >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 3 }-> or(0, 0) :|: z = 1 + x', z' = 1 + 0 + 0, x' >= 0 member(z, z') -{ 3 }-> or(0, 0) :|: z' = 1 + (1 + y') + 0, y' >= 0, z = 0 member(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 or(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x sort(z) -{ 1 }-> st(0, l) :|: z = l, l >= 0 st(z, z') -{ 2 }-> cond1(or(equal(n, m'), member(n, l')), n, 1 + m' + l') :|: n >= 0, z = n, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, n, 0) :|: n >= 0, z = n, z' = 0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x or(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 if(z, z', z'') -{ 1 }-> u :|: z'' = v, v >= 0, z = 1, z' = u, u >= 0 if(z, z', z'') -{ 1 }-> v :|: z'' = v, v >= 0, z' = u, z = 0, u >= 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(n, if(gt(u', max(l'')), u', max(l''))), n, 1 + u' + l'') :|: n >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z' = n, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(n, 0), n, 0) :|: z'' = 0, n >= 0, z' = n, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + n + st(1 + n, l) :|: n >= 0, z = 1, z' = n, l >= 0, z'' = l cond2(z, z', z'') -{ 1 }-> st(1 + n, l) :|: n >= 0, z' = n, l >= 0, z = 0, z'' = l cond2(z, z', z'') -{ 1 }-> 0 :|: n >= 0, z = 1, z' = n, l >= 0, z'' = l equal(z, z') -{ 1 }-> equal(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0 gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0 if(z, z', z'') -{ 1 }-> u :|: z'' = v, v >= 0, z = 1, z' = u, u >= 0 if(z, z', z'') -{ 1 }-> v :|: z'' = v, v >= 0, z' = u, z = 0, u >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(u, 0), u, 0) :|: z = 1 + u + 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z = 1 + x', z' = 1 + 0 + 0, x' >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + (1 + y') + 0, y' >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(x'', y''), or(equal(1 + x'', m3), member(1 + x'', l4))) :|: z = 1 + x'', y'' >= 0, x'' >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(x'', y''), 0) :|: z = 1 + x'', z' = 1 + (1 + y'') + 0, y'' >= 0, x'' >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + x', m1), member(1 + x', l2))) :|: z = 1 + x', x' >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 or(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x sort(z) -{ 1 }-> st(0, l) :|: z = l, l >= 0 st(z, z') -{ 2 }-> cond1(or(equal(n, m'), member(n, l')), n, 1 + m' + l') :|: n >= 0, z = n, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, n, 0) :|: n >= 0, z = n, z' = 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', 0), z', 0) :|: z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 1 }-> gt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(z - 1, 0), z - 1, 0) :|: z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { gt } { equal } { if } { or } { max } { member } { cond1, cond2, st } { sort } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', 0), z', 0) :|: z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 1 }-> gt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(z - 1, 0), z - 1, 0) :|: z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {gt}, {equal}, {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', 0), z', 0) :|: z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 1 }-> gt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(z - 1, 0), z - 1, 0) :|: z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {gt}, {equal}, {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: gt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', 0), z', 0) :|: z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 1 }-> gt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(z - 1, 0), z - 1, 0) :|: z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {gt}, {equal}, {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: gt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', 0), z', 0) :|: z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 1 }-> gt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 3 }-> if(gt(z - 1, 0), z - 1, 0) :|: z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {equal}, {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 5 }-> if(s'', z - 1, 0) :|: s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {equal}, {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 5 }-> if(s'', z - 1, 0) :|: s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {equal}, {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: equal after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 1 }-> equal(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 5 }-> if(s'', z - 1, 0) :|: s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 3 }-> or(equal(z - 1, y''), or(equal(1 + (z - 1), m3), member(1 + (z - 1), l4))) :|: y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 3 }-> or(equal(z - 1, z' - 2), 0) :|: z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 3 }-> or(1, or(equal(0, m''), member(0, l1))) :|: m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 3 }-> or(0, or(equal(0, m2), member(0, l3))) :|: y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 3 }-> or(0, or(equal(1 + (z - 1), m1), member(1 + (z - 1), l2))) :|: z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 2 }-> cond1(or(equal(z, m'), member(z, l')), z, 1 + m' + l') :|: z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 5 }-> if(s'', z - 1, 0) :|: s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 5 + z }-> or(s6, 0) :|: s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 5 }-> if(s'', z - 1, 0) :|: s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 5 + z }-> or(s6, 0) :|: s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {if}, {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 5 }-> if(s'', z - 1, 0) :|: s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 5 + z }-> or(s6, 0) :|: s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 5 + z }-> or(s6, 0) :|: s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 5 + z }-> or(s6, 0) :|: s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {or}, {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 5 + z }-> or(s6, 0) :|: s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {max}, {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: ?, size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: max after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 2 }-> cond2(gt(z', if(gt(u', max(l'')), u', max(l''))), z', 1 + u' + l'') :|: z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 3 }-> if(gt(u, if(gt(u'', max(l5)), u'', max(l5))), u, if(gt(u'', max(l5)), u'', max(l5))) :|: u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 7 + inf + inf' + s11 + s14 }-> cond2(s15, z', 1 + u' + l'') :|: s11 >= 0, s11 <= l'', s12 >= 0, s12 <= 1, s13 >= 0, s13 <= l'', s14 >= 0, s14 <= s13 + u', s15 >= 0, s15 <= 1, z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 12 + inf'' + inf1 + inf2 + inf3 + s16 + s19 + s21 }-> s25 :|: s16 >= 0, s16 <= l5, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= l5, s19 >= 0, s19 <= s18 + u'', s20 >= 0, s20 <= 1, s21 >= 0, s21 <= l5, s22 >= 0, s22 <= 1, s23 >= 0, s23 <= l5, s24 >= 0, s24 <= s23 + u'', s25 >= 0, s25 <= s24 + u, u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: member after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 7 + inf + inf' + s11 + s14 }-> cond2(s15, z', 1 + u' + l'') :|: s11 >= 0, s11 <= l'', s12 >= 0, s12 <= 1, s13 >= 0, s13 <= l'', s14 >= 0, s14 <= s13 + u', s15 >= 0, s15 <= 1, z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 12 + inf'' + inf1 + inf2 + inf3 + s16 + s19 + s21 }-> s25 :|: s16 >= 0, s16 <= l5, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= l5, s19 >= 0, s19 <= s18 + u'', s20 >= 0, s20 <= 1, s21 >= 0, s21 <= l5, s22 >= 0, s22 <= 1, s23 >= 0, s23 <= l5, s24 >= 0, s24 <= s23 + u'', s25 >= 0, s25 <= s24 + u, u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {member}, {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] member: runtime: ?, size: O(1) [2] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: member after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 19 + z + 3*z*z' + 34*z' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 7 + inf + inf' + s11 + s14 }-> cond2(s15, z', 1 + u' + l'') :|: s11 >= 0, s11 <= l'', s12 >= 0, s12 <= 1, s13 >= 0, s13 <= l'', s14 >= 0, s14 <= s13 + u', s15 >= 0, s15 <= 1, z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 12 + inf'' + inf1 + inf2 + inf3 + s16 + s19 + s21 }-> s25 :|: s16 >= 0, s16 <= l5, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= l5, s19 >= 0, s19 <= s18 + u'', s20 >= 0, s20 <= 1, s21 >= 0, s21 <= l5, s22 >= 0, s22 <= 1, s23 >= 0, s23 <= l5, s24 >= 0, s24 <= s23 + u'', s25 >= 0, s25 <= s24 + u, u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 8 + 2*z }-> or(s7, or(s8, member(1 + (z - 1), l4))) :|: s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 6 }-> or(1, or(s3, member(0, l1))) :|: s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 6 + z }-> or(0, or(s4, member(1 + (z - 1), l2))) :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 6 }-> or(0, or(s5, member(0, l3))) :|: s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 5 + z }-> cond1(or(s2, member(z, l')), z, 1 + m' + l') :|: s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] member: runtime: O(n^2) [19 + z + 3*z*z' + 34*z'], size: O(1) [2] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 7 + inf + inf' + s11 + s14 }-> cond2(s15, z', 1 + u' + l'') :|: s11 >= 0, s11 <= l'', s12 >= 0, s12 <= 1, s13 >= 0, s13 <= l'', s14 >= 0, s14 <= s13 + u', s15 >= 0, s15 <= 1, z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 12 + inf'' + inf1 + inf2 + inf3 + s16 + s19 + s21 }-> s25 :|: s16 >= 0, s16 <= l5, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= l5, s19 >= 0, s19 <= s18 + u'', s20 >= 0, s20 <= 1, s21 >= 0, s21 <= l5, s22 >= 0, s22 <= 1, s23 >= 0, s23 <= l5, s24 >= 0, s24 <= s23 + u'', s25 >= 0, s25 <= s24 + u, u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 27 + 34*l1 }-> s30 :|: s28 >= 0, s28 <= 2, s29 >= 0, s29 <= s3 + 1, s30 >= 0, s30 <= 1 + 1, s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 27 + 34*l2 + 3*l2*z + 2*z }-> s33 :|: s31 >= 0, s31 <= 2, s32 >= 0, s32 <= s4 + 1, s33 >= 0, s33 <= 0 + 1, s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 27 + 34*l3 }-> s36 :|: s34 >= 0, s34 <= 2, s35 >= 0, s35 <= s5 + 1, s36 >= 0, s36 <= 0 + 1, s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 29 + 34*l4 + 3*l4*z + 3*z }-> s39 :|: s37 >= 0, s37 <= 2, s38 >= 0, s38 <= s8 + 1, s39 >= 0, s39 <= s7 + 1, s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 25 + 34*l' + 3*l'*z + 2*z }-> cond1(s27, z, 1 + m' + l') :|: s26 >= 0, s26 <= 2, s27 >= 0, s27 <= s2 + 1, s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] member: runtime: O(n^2) [19 + z + 3*z*z' + 34*z'], size: O(1) [2] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: cond1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: cond2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: st after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 7 + inf + inf' + s11 + s14 }-> cond2(s15, z', 1 + u' + l'') :|: s11 >= 0, s11 <= l'', s12 >= 0, s12 <= 1, s13 >= 0, s13 <= l'', s14 >= 0, s14 <= s13 + u', s15 >= 0, s15 <= 1, z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 12 + inf'' + inf1 + inf2 + inf3 + s16 + s19 + s21 }-> s25 :|: s16 >= 0, s16 <= l5, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= l5, s19 >= 0, s19 <= s18 + u'', s20 >= 0, s20 <= 1, s21 >= 0, s21 <= l5, s22 >= 0, s22 <= 1, s23 >= 0, s23 <= l5, s24 >= 0, s24 <= s23 + u'', s25 >= 0, s25 <= s24 + u, u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 27 + 34*l1 }-> s30 :|: s28 >= 0, s28 <= 2, s29 >= 0, s29 <= s3 + 1, s30 >= 0, s30 <= 1 + 1, s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 27 + 34*l2 + 3*l2*z + 2*z }-> s33 :|: s31 >= 0, s31 <= 2, s32 >= 0, s32 <= s4 + 1, s33 >= 0, s33 <= 0 + 1, s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 27 + 34*l3 }-> s36 :|: s34 >= 0, s34 <= 2, s35 >= 0, s35 <= s5 + 1, s36 >= 0, s36 <= 0 + 1, s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 29 + 34*l4 + 3*l4*z + 3*z }-> s39 :|: s37 >= 0, s37 <= 2, s38 >= 0, s38 <= s8 + 1, s39 >= 0, s39 <= s7 + 1, s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 25 + 34*l' + 3*l'*z + 2*z }-> cond1(s27, z, 1 + m' + l') :|: s26 >= 0, s26 <= 2, s27 >= 0, s27 <= s2 + 1, s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] member: runtime: O(n^2) [19 + z + 3*z*z' + 34*z'], size: O(1) [2] cond1: runtime: ?, size: INF cond2: runtime: ?, size: INF st: runtime: ?, size: INF ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cond1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 4 }-> cond2(s, z', 0) :|: s >= 0, s <= 1, z'' = 0, z' >= 0, z = 0 cond1(z, z', z'') -{ 7 + inf + inf' + s11 + s14 }-> cond2(s15, z', 1 + u' + l'') :|: s11 >= 0, s11 <= l'', s12 >= 0, s12 <= 1, s13 >= 0, s13 <= l'', s14 >= 0, s14 <= s13 + u', s15 >= 0, s15 <= 1, z' >= 0, l'' >= 0, z'' = 1 + u' + l'', u' >= 0, z = 0 cond1(z, z', z'') -{ 1 }-> 1 + z' + st(1 + z', z'') :|: z' >= 0, z = 1, z'' >= 0 cond2(z, z', z'') -{ 1 }-> st(1 + z', z'') :|: z' >= 0, z'' >= 0, z = 0 cond2(z, z', z'') -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0 equal(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 gt(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 1 :|: z' = 0, z - 1 >= 0 gt(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' >= 0 max(z) -{ 12 + inf'' + inf1 + inf2 + inf3 + s16 + s19 + s21 }-> s25 :|: s16 >= 0, s16 <= l5, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= l5, s19 >= 0, s19 <= s18 + u'', s20 >= 0, s20 <= 1, s21 >= 0, s21 <= l5, s22 >= 0, s22 <= 1, s23 >= 0, s23 <= l5, s24 >= 0, s24 <= s23 + u'', s25 >= 0, s25 <= s24 + u, u'' >= 0, z = 1 + u + (1 + u'' + l5), l5 >= 0, u >= 0 max(z) -{ 6 }-> s9 :|: s9 >= 0, s9 <= 0 + (z - 1), s'' >= 0, s'' <= 1, z - 1 >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 6 + z }-> s10 :|: s10 >= 0, s10 <= s6 + 1, s6 >= 0, s6 <= 1, z' - 2 >= 0, z - 1 >= 0 member(z, z') -{ 27 + 34*l1 }-> s30 :|: s28 >= 0, s28 <= 2, s29 >= 0, s29 <= s3 + 1, s30 >= 0, s30 <= 1 + 1, s3 >= 0, s3 <= 1, m'' >= 0, z' = 1 + 0 + (1 + m'' + l1), l1 >= 0, z = 0 member(z, z') -{ 27 + 34*l2 + 3*l2*z + 2*z }-> s33 :|: s31 >= 0, s31 <= 2, s32 >= 0, s32 <= s4 + 1, s33 >= 0, s33 <= 0 + 1, s4 >= 0, s4 <= 1, z - 1 >= 0, z' = 1 + 0 + (1 + m1 + l2), m1 >= 0, l2 >= 0 member(z, z') -{ 27 + 34*l3 }-> s36 :|: s34 >= 0, s34 <= 2, s35 >= 0, s35 <= s5 + 1, s36 >= 0, s36 <= 0 + 1, s5 >= 0, s5 <= 1, y' >= 0, z' = 1 + (1 + y') + (1 + m2 + l3), z = 0, m2 >= 0, l3 >= 0 member(z, z') -{ 29 + 34*l4 + 3*l4*z + 3*z }-> s39 :|: s37 >= 0, s37 <= 2, s38 >= 0, s38 <= s8 + 1, s39 >= 0, s39 <= s7 + 1, s7 >= 0, s7 <= 1, s8 >= 0, s8 <= 1, y'' >= 0, z - 1 >= 0, z' = 1 + (1 + y'') + (1 + m3 + l4), l4 >= 0, m3 >= 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z = 0, x >= 0, 1 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' = 1 + 0 + 0, z - 1 >= 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 4 }-> x :|: z' - 2 >= 0, z = 0, x >= 0, 0 = x, 0 = 0 member(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 sort(z) -{ 1 }-> st(0, z) :|: z >= 0 st(z, z') -{ 25 + 34*l' + 3*l'*z + 2*z }-> cond1(s27, z, 1 + m' + l') :|: s26 >= 0, s26 <= 2, s27 >= 0, s27 <= s2 + 1, s2 >= 0, s2 <= 1, z >= 0, z' = 1 + m' + l', m' >= 0, l' >= 0 st(z, z') -{ 2 }-> cond1(0, z, 0) :|: z >= 0, z' = 0 Function symbols to be analyzed: {cond1,cond2,st}, {sort} Previous analysis results are: gt: runtime: O(n^1) [2 + z'], size: O(1) [1] equal: runtime: O(n^1) [3 + z], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] or: runtime: O(1) [1], size: O(n^1) [1 + z] max: runtime: EXP, size: O(n^1) [z] member: runtime: O(n^2) [19 + z + 3*z*z' + 34*z'], size: O(1) [2] cond1: runtime: INF, size: INF cond2: runtime: ?, size: INF st: runtime: ?, size: INF ---------------------------------------- (63) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (64) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sort(l) -> st(0, l) [1] st(n, l) -> cond1(member(n, l), n, l) [1] cond1(true, n, l) -> cons(n, st(s(n), l)) [1] cond1(false, n, l) -> cond2(gt(n, max(l)), n, l) [1] cond2(true, n, l) -> nil [1] cond2(false, n, l) -> st(s(n), l) [1] member(n, nil) -> false [1] member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) [1] or(x, true) -> true [1] or(x, false) -> x [1] equal(0, 0) -> true [1] equal(s(x), 0) -> false [1] equal(0, s(y)) -> false [1] equal(s(x), s(y)) -> equal(x, y) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] max(nil) -> 0 [1] max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) [1] if(true, u, v) -> u [1] if(false, u, v) -> v [1] The TRS has the following type information: sort :: cons:nil -> cons:nil st :: 0:s -> cons:nil -> cons:nil 0 :: 0:s cond1 :: true:false -> 0:s -> cons:nil -> cons:nil member :: 0:s -> cons:nil -> true:false true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false cond2 :: true:false -> 0:s -> cons:nil -> cons:nil gt :: 0:s -> 0:s -> true:false max :: cons:nil -> 0:s nil :: cons:nil or :: true:false -> true:false -> true:false equal :: 0:s -> 0:s -> true:false if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (65) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 1 }-> cond2(gt(n, max(l)), n, l) :|: n >= 0, z' = n, l >= 0, z = 0, z'' = l cond1(z, z', z'') -{ 1 }-> 1 + n + st(1 + n, l) :|: n >= 0, z = 1, z' = n, l >= 0, z'' = l cond2(z, z', z'') -{ 1 }-> st(1 + n, l) :|: n >= 0, z' = n, l >= 0, z = 0, z'' = l cond2(z, z', z'') -{ 1 }-> 0 :|: n >= 0, z = 1, z' = n, l >= 0, z'' = l equal(z, z') -{ 1 }-> equal(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 equal(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 equal(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0 gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0 if(z, z', z'') -{ 1 }-> u :|: z'' = v, v >= 0, z = 1, z' = u, u >= 0 if(z, z', z'') -{ 1 }-> v :|: z'' = v, v >= 0, z' = u, z = 0, u >= 0 max(z) -{ 1 }-> if(gt(u, max(l)), u, max(l)) :|: z = 1 + u + l, l >= 0, u >= 0 max(z) -{ 1 }-> 0 :|: z = 0 member(z, z') -{ 1 }-> or(equal(n, m), member(n, l)) :|: n >= 0, z = n, l >= 0, z' = 1 + m + l, m >= 0 member(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 or(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x sort(z) -{ 1 }-> st(0, l) :|: z = l, l >= 0 st(z, z') -{ 1 }-> cond1(member(n, l), n, l) :|: z' = l, n >= 0, z = n, l >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (67) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: sort(z0) -> st(0, z0) st(z0, z1) -> cond1(member(z0, z1), z0, z1) cond1(true, z0, z1) -> cons(z0, st(s(z0), z1)) cond1(false, z0, z1) -> cond2(gt(z0, max(z1)), z0, z1) cond2(true, z0, z1) -> nil cond2(false, z0, z1) -> st(s(z0), z1) member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: SORT(z0) -> c(ST(0, z0)) ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(true, z0, z1) -> c4 COND2(false, z0, z1) -> c5(ST(s(z0), z1)) MEMBER(z0, nil) -> c6 MEMBER(z0, cons(z1, z2)) -> c7(OR(equal(z0, z1), member(z0, z2)), EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(OR(equal(z0, z1), member(z0, z2)), MEMBER(z0, z2)) OR(z0, true) -> c9 OR(z0, false) -> c10 EQUAL(0, 0) -> c11 EQUAL(s(z0), 0) -> c12 EQUAL(0, s(z0)) -> c13 EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(0, z0) -> c15 GT(s(z0), 0) -> c16 GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MAX(nil) -> c18 MAX(cons(z0, z1)) -> c19(IF(gt(z0, max(z1)), z0, max(z1)), GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(IF(gt(z0, max(z1)), z0, max(z1)), MAX(z1)) IF(true, z0, z1) -> c21 IF(false, z0, z1) -> c22 S tuples: SORT(z0) -> c(ST(0, z0)) ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(true, z0, z1) -> c4 COND2(false, z0, z1) -> c5(ST(s(z0), z1)) MEMBER(z0, nil) -> c6 MEMBER(z0, cons(z1, z2)) -> c7(OR(equal(z0, z1), member(z0, z2)), EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(OR(equal(z0, z1), member(z0, z2)), MEMBER(z0, z2)) OR(z0, true) -> c9 OR(z0, false) -> c10 EQUAL(0, 0) -> c11 EQUAL(s(z0), 0) -> c12 EQUAL(0, s(z0)) -> c13 EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(0, z0) -> c15 GT(s(z0), 0) -> c16 GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MAX(nil) -> c18 MAX(cons(z0, z1)) -> c19(IF(gt(z0, max(z1)), z0, max(z1)), GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(IF(gt(z0, max(z1)), z0, max(z1)), MAX(z1)) IF(true, z0, z1) -> c21 IF(false, z0, z1) -> c22 K tuples:none Defined Rule Symbols: sort_1, st_2, cond1_3, cond2_3, member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: SORT_1, ST_2, COND1_3, COND2_3, MEMBER_2, OR_2, EQUAL_2, GT_2, MAX_1, IF_3 Compound Symbols: c_1, c1_2, c2_1, c3_3, c4, c5_1, c6, c7_2, c8_2, c9, c10, c11, c12, c13, c14_1, c15, c16, c17_1, c18, c19_3, c20_2, c21, c22 ---------------------------------------- (69) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: SORT(z0) -> c(ST(0, z0)) Removed 12 trailing nodes: EQUAL(s(z0), 0) -> c12 GT(0, z0) -> c15 MEMBER(z0, nil) -> c6 GT(s(z0), 0) -> c16 IF(false, z0, z1) -> c22 EQUAL(0, 0) -> c11 IF(true, z0, z1) -> c21 MAX(nil) -> c18 COND2(true, z0, z1) -> c4 EQUAL(0, s(z0)) -> c13 OR(z0, false) -> c10 OR(z0, true) -> c9 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: sort(z0) -> st(0, z0) st(z0, z1) -> cond1(member(z0, z1), z0, z1) cond1(true, z0, z1) -> cons(z0, st(s(z0), z1)) cond1(false, z0, z1) -> cond2(gt(z0, max(z1)), z0, z1) cond2(true, z0, z1) -> nil cond2(false, z0, z1) -> st(s(z0), z1) member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) MEMBER(z0, cons(z1, z2)) -> c7(OR(equal(z0, z1), member(z0, z2)), EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(OR(equal(z0, z1), member(z0, z2)), MEMBER(z0, z2)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MAX(cons(z0, z1)) -> c19(IF(gt(z0, max(z1)), z0, max(z1)), GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(IF(gt(z0, max(z1)), z0, max(z1)), MAX(z1)) S tuples: ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) MEMBER(z0, cons(z1, z2)) -> c7(OR(equal(z0, z1), member(z0, z2)), EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(OR(equal(z0, z1), member(z0, z2)), MEMBER(z0, z2)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MAX(cons(z0, z1)) -> c19(IF(gt(z0, max(z1)), z0, max(z1)), GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(IF(gt(z0, max(z1)), z0, max(z1)), MAX(z1)) K tuples:none Defined Rule Symbols: sort_1, st_2, cond1_3, cond2_3, member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: ST_2, COND1_3, COND2_3, MEMBER_2, EQUAL_2, GT_2, MAX_1 Compound Symbols: c1_2, c2_1, c3_3, c5_1, c7_2, c8_2, c14_1, c17_1, c19_3, c20_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: sort(z0) -> st(0, z0) st(z0, z1) -> cond1(member(z0, z1), z0, z1) cond1(true, z0, z1) -> cons(z0, st(s(z0), z1)) cond1(false, z0, z1) -> cond2(gt(z0, max(z1)), z0, z1) cond2(true, z0, z1) -> nil cond2(false, z0, z1) -> st(s(z0), z1) member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) S tuples: ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) K tuples:none Defined Rule Symbols: sort_1, st_2, cond1_3, cond2_3, member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: ST_2, COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1 Compound Symbols: c1_2, c2_1, c3_3, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1 ---------------------------------------- (73) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: sort(z0) -> st(0, z0) st(z0, z1) -> cond1(member(z0, z1), z0, z1) cond1(true, z0, z1) -> cons(z0, st(s(z0), z1)) cond1(false, z0, z1) -> cond2(gt(z0, max(z1)), z0, z1) cond2(true, z0, z1) -> nil cond2(false, z0, z1) -> st(s(z0), z1) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) S tuples: ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) K tuples:none Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: ST_2, COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1 Compound Symbols: c1_2, c2_1, c3_3, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ST(z0, z1) -> c1(COND1(member(z0, z1), z0, z1), MEMBER(z0, z1)) by ST(z0, nil) -> c1(COND1(false, z0, nil), MEMBER(z0, nil)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil), MEMBER(z0, nil)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil), MEMBER(z0, nil)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) K tuples:none Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c3_3, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1, c1_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) K tuples:none Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c3_3, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1, c1_2, c1_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1(false, z0, z1) -> c3(COND2(gt(z0, max(z1)), z0, z1), GT(z0, max(z1)), MAX(z1)) by COND1(false, 0, x1) -> c3(COND2(false, 0, x1), GT(0, max(x1)), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil)), MAX(nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), GT(0, max(x1)), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil)), MAX(nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), GT(0, max(x1)), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil)), MAX(nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) K tuples:none Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1, c1_2, c1_1, c3_3 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) K tuples:none Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1, c1_2, c1_1, c3_3, c3_2 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) We considered the (Usable) Rules:none And the Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_2 POL(COND2(x_1, x_2, x_3)) = 0 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c19_2, c20_1, c1_2, c1_1, c3_3, c3_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MAX(cons(z0, z1)) -> c19(GT(z0, max(z1)), MAX(z1)) by MAX(cons(x0, nil)) -> c19(GT(x0, 0), MAX(nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, nil)) -> c19(GT(x0, 0), MAX(nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, nil)) -> c19(GT(x0, 0), MAX(nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c3_3, c3_2, c19_2 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MAX(cons(x0, nil)) -> c19(GT(x0, 0), MAX(nil)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c3_3, c3_2, c19_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ST(z0, cons(z1, z2)) -> c1(COND1(or(equal(z0, z1), member(z0, z2)), z0, cons(z1, z2)), MEMBER(z0, cons(z1, z2))) by ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(0, cons(0, x2)) -> c1(COND1(or(true, member(0, x2)), 0, cons(0, x2)), MEMBER(0, cons(0, x2))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(s(z0), x2)) -> c1(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2)), MEMBER(0, cons(s(z0), x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(0, cons(0, x2)) -> c1(COND1(or(true, member(0, x2)), 0, cons(0, x2)), MEMBER(0, cons(0, x2))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(s(z0), x2)) -> c1(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2)), MEMBER(0, cons(s(z0), x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(0, cons(0, x2)) -> c1(COND1(or(true, member(0, x2)), 0, cons(0, x2)), MEMBER(0, cons(0, x2))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(s(z0), x2)) -> c1(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2)), MEMBER(0, cons(s(z0), x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_3, c3_2, c19_2, c1_2 ---------------------------------------- (91) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(0, x2)) -> c(MEMBER(0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(0, cons(s(z0), x2)) -> c(MEMBER(0, cons(s(z0), x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(0, x2)) -> c(MEMBER(0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(0, cons(s(z0), x2)) -> c(MEMBER(0, cons(s(z0), x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_3, c3_2, c19_2, c1_2, c_1 ---------------------------------------- (93) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ST(0, cons(0, x2)) -> c(MEMBER(0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(MEMBER(0, cons(s(z0), x2))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_3, c3_2, c19_2, c1_2, c_1 ---------------------------------------- (95) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_3, c3_2, c19_2, c1_2, c_1 ---------------------------------------- (97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) We considered the (Usable) Rules:none And the Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1(x_1, x_2, x_3)) = x_3 POL(COND2(x_1, x_2, x_3)) = x_3 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_3, c3_2, c19_2, c1_2, c_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1(false, x0, cons(z0, z1)) -> c3(COND2(gt(x0, if(gt(z0, max(z1)), z0, max(z1))), x0, cons(z0, z1)), GT(x0, max(cons(z0, z1))), MAX(cons(z0, z1))) by COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), GT(0, max(cons(x1, x2))), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c3(GT(x0, max(cons(x1, x2))), MAX(cons(x1, x2))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), GT(0, max(cons(x1, x2))), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c3(GT(x0, max(cons(x1, x2))), MAX(cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), GT(0, max(cons(x1, x2))), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c3(GT(x0, max(cons(x1, x2))), MAX(cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c3(GT(x0, max(cons(x1, x2))), MAX(cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c3(GT(x0, max(cons(x1, x2))), MAX(cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3 ---------------------------------------- (103) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) We considered the (Usable) Rules:none And the Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_2 POL(COND2(x_1, x_2, x_3)) = 0 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) We considered the (Usable) Rules:none And the Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_3 POL(COND2(x_1, x_2, x_3)) = x_3 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = [1] POL(true) = 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1(false, x0, nil) -> c3(COND2(gt(x0, 0), x0, nil), GT(x0, max(nil))) by COND1(false, 0, nil) -> c3(COND2(false, 0, nil), GT(0, max(nil))) COND1(false, s(z0), nil) -> c3(COND2(true, s(z0), nil), GT(s(z0), max(nil))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil), GT(0, max(nil))) COND1(false, s(z0), nil) -> c3(COND2(true, s(z0), nil), GT(s(z0), max(nil))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil), GT(0, max(nil))) COND1(false, s(z0), nil) -> c3(COND2(true, s(z0), nil), GT(s(z0), max(nil))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1, c3_1 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1, c3_1 ---------------------------------------- (113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) We considered the (Usable) Rules:none And the Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_2 POL(COND2(x_1, x_2, x_3)) = 0 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1, c3_1 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) We considered the (Usable) Rules:none And the Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1(x_1, x_2, x_3)) = [1] POL(COND2(x_1, x_2, x_3)) = [1] POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c19_2, c1_2, c_1, c3_3, c4_1, c3_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MAX(cons(x0, cons(z0, z1))) -> c19(GT(x0, if(gt(z0, max(z1)), z0, max(z1))), MAX(cons(z0, z1))) by MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ST(z0, cons(x1, nil)) -> c1(COND1(or(equal(z0, x1), false), z0, cons(x1, nil)), MEMBER(z0, cons(x1, nil))) by ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) S tuples: COND1(true, z0, z1) -> c2(ST(s(z0), z1)) COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND1_3, COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2 Compound Symbols: c2_1, c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1 ---------------------------------------- (121) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND1(true, z0, z1) -> c2(ST(s(z0), z1)) by COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) S tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3 Compound Symbols: c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) We considered the (Usable) Rules:none And the Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_2 POL(COND2(x_1, x_2, x_3)) = 0 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) S tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3 Compound Symbols: c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1 ---------------------------------------- (125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ST(z0, nil) -> c1(COND1(false, z0, nil)) We considered the (Usable) Rules: gt(0, z0) -> false gt(s(z0), s(z1)) -> gt(z0, z1) gt(s(z0), 0) -> true And the Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_2 + x_3 POL(COND2(x_1, x_2, x_3)) = x_1 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(equal(x_1, x_2)) = 0 POL(false) = [1] POL(gt(x_1, x_2)) = [1] POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) S tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3 Compound Symbols: c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, max(nil)), x1, 0)), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) by COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) S tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3 Compound Symbols: c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) by COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) S tuples: COND2(false, z0, z1) -> c5(ST(s(z0), z1)) EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: COND2_3, EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3 Compound Symbols: c5_1, c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1 ---------------------------------------- (131) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND2(false, z0, z1) -> c5(ST(s(z0), z1)) by COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (133) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: COND2(false, 0, nil) -> c5(ST(s(0), nil)) ST(z0, nil) -> c1(COND1(false, z0, nil)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND2(false, 0, nil) -> c5(ST(s(0), nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) We considered the (Usable) Rules:none And the Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(COND1(x_1, x_2, x_3)) = x_2 POL(COND2(x_1, x_2, x_3)) = x_2 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = 0 POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, cons(0, x2)) -> c3(COND2(gt(x0, if(false, 0, max(x2))), x0, cons(0, x2)), GT(x0, max(cons(0, x2))), MAX(cons(0, x2))) by COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) ST(z0, nil) -> c1(COND1(false, z0, nil)) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_1, c3_2, c1_2, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace ST(z0, nil) -> c1(COND1(false, z0, nil)) by ST(s(0), nil) -> c1(COND1(false, s(0), nil)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, COND1_3, ST_2, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c3_2, c1_2, c1_1, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (141) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND1(false, 0, nil) -> c3(COND2(false, 0, nil)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) K tuples: COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, COND1_3, ST_2, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c3_2, c1_2, c1_1, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (143) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND1(false, 0, x1) -> c3(COND2(false, 0, x1), MAX(x1)) by COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, nil) -> c5(ST(s(0), nil)) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (145) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2(false, 0, nil) -> c5(ST(s(0), nil)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, cons(x1, nil)) -> c3(COND2(gt(x0, if(gt(x1, 0), x1, max(nil))), x0, cons(x1, nil)), GT(x0, max(cons(x1, nil))), MAX(cons(x1, nil))) by COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace ST(z0, cons(x1, cons(z1, z2))) -> c1(COND1(or(equal(z0, x1), or(equal(z0, z1), member(z0, z2))), z0, cons(x1, cons(z1, z2))), MEMBER(z0, cons(x1, cons(z1, z2)))) by ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c3(COND2(false, 0, cons(x1, cons(x2, x3))), MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1 ---------------------------------------- (151) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1, c6_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ST(z0, cons(z1, nil)) -> c1(COND1(equal(z0, z1), z0, cons(z1, nil)), MEMBER(z0, cons(z1, nil))) by ST(0, cons(0, nil)) -> c1(COND1(true, 0, cons(0, nil)), MEMBER(0, cons(0, nil))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(0, cons(s(z0), nil)) -> c1(COND1(false, 0, cons(s(z0), nil)), MEMBER(0, cons(s(z0), nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(0, cons(0, nil)) -> c1(COND1(true, 0, cons(0, nil)), MEMBER(0, cons(0, nil))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(0, cons(s(z0), nil)) -> c1(COND1(false, 0, cons(s(z0), nil)), MEMBER(0, cons(s(z0), nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(0, cons(0, nil)) -> c1(COND1(true, 0, cons(0, nil)), MEMBER(0, cons(0, nil))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(0, cons(s(z0), nil)) -> c1(COND1(false, 0, cons(s(z0), nil)), MEMBER(0, cons(s(z0), nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1, c6_1 ---------------------------------------- (155) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ST(0, cons(s(z0), nil)) -> c1(COND1(false, 0, cons(s(z0), nil)), MEMBER(0, cons(s(z0), nil))) ST(0, cons(0, nil)) -> c1(COND1(true, 0, cons(0, nil)), MEMBER(0, cons(0, nil))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c3(COND2(false, 0, cons(x1, x2)), MAX(cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(0, x0)) -> c3(COND2(false, 0, cons(0, x0)), MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c3(COND2(false, 0, cons(s(x0), x1)), MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c3(COND2(false, 0, cons(x1, nil)), MAX(cons(x1, nil))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c3_2, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1, c6_1 ---------------------------------------- (157) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1, c6_1, c9_1 ---------------------------------------- (159) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) We considered the (Usable) Rules:none And the Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1(x_1, x_2, x_3)) = x_3 POL(COND2(x_1, x_2, x_3)) = x_3 POL(EQUAL(x_1, x_2)) = 0 POL(GT(x_1, x_2)) = 0 POL(MAX(x_1)) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(ST(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(equal(x_1, x_2)) = 0 POL(false) = 0 POL(gt(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] POL(max(x_1)) = 0 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(or(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c3_3, c4_1, c3_1, c19_2, c19_1, c2_1, c5_1, c6_1, c9_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, cons(x1, cons(z0, z1))) -> c3(COND2(gt(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), x0, cons(x1, cons(z0, z1))), GT(x0, max(cons(x1, cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) by COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c9_1 ---------------------------------------- (163) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace ST(s(z0), cons(s(z1), x2)) -> c1(COND1(or(equal(z0, z1), member(s(z0), x2)), s(z0), cons(s(z1), x2)), MEMBER(s(z0), cons(s(z1), x2))) by ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) S tuples: EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: EQUAL_2, GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3 Compound Symbols: c14_1, c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c9_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQUAL(s(z0), s(z1)) -> c14(EQUAL(z0, z1)) by EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c9_1, c14_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) by COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c9_1, c14_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, x0, nil) -> c3(GT(x0, max(nil))) by COND1(false, z0, nil) -> c3(GT(z0, 0)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) COND1(false, z0, nil) -> c3(GT(z0, 0)) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(GT(x0, max(cons(x1, x2)))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, x0, nil) -> c3(GT(x0, max(nil))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c9_1, c14_1 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND1(false, z0, nil) -> c3(GT(z0, 0)) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c1_1, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c9_1, c14_1 ---------------------------------------- (173) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace ST(x0, cons(x1, x2)) -> c1(MEMBER(x0, cons(x1, x2))) by ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c3_1, c19_2, c19_1, c2_1, c3_3, c5_1, c1_1, c6_1, c9_1, c14_1 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) by COND1(false, s(z0), nil) -> c3(GT(s(z0), 0)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) COND1(false, s(z0), nil) -> c3(GT(s(z0), 0)) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(false, s(z0), nil) -> c3(GT(s(z0), max(nil))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_2, c19_1, c2_1, c3_3, c5_1, c1_1, c6_1, c9_1, c14_1, c3_1 ---------------------------------------- (177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: COND1(false, s(z0), nil) -> c3(GT(s(z0), 0)) ST(s(0), nil) -> c1(COND1(false, s(0), nil)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, max(nil)), x1, 0)), MAX(cons(x1, nil))) by MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, max(cons(z0, z1))), x1, if(gt(z0, max(z1)), z0, max(z1)))), MAX(cons(x1, cons(z0, z1)))) by MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MAX(cons(x0, cons(0, x2))) -> c19(GT(x0, if(false, 0, max(x2))), MAX(cons(0, x2))) by MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MAX(cons(x0, cons(x1, nil))) -> c19(GT(x0, if(gt(x1, 0), x1, max(nil))), MAX(cons(x1, nil))) by MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_2, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MAX(cons(x0, cons(x1, cons(z0, z1)))) -> c19(GT(x0, if(gt(x1, if(gt(z0, max(z1)), z0, max(z1))), x1, max(cons(z0, z1)))), MAX(cons(x1, cons(z0, z1)))) by MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1, c19_2 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) by COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c3_3, c5_1, c6_1, c1_1, c9_1, c14_1, c19_2 ---------------------------------------- (191) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) by COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c3_3, c6_1, c1_1, c9_1, c14_1, c19_2 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(gt(0, max(z1)), 0, max(z1))), MAX(cons(0, z1))) by COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c3_3, c6_1, c1_1, c9_1, c14_1, c19_2 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, max(nil))), MAX(cons(z1, nil))) by COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) S tuples: GT(s(z0), s(z1)) -> c17(GT(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: GT_2, MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2 Compound Symbols: c17_1, c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c19_2 ---------------------------------------- (197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GT(s(z0), s(z1)) -> c17(GT(z0, z1)) by GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c19_2, c17_1 ---------------------------------------- (199) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ST(s(s(x0)), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), 0), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(0, cons(z2, z3))), MEMBER(s(s(x0)), cons(0, cons(z2, z3)))) by ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c19_2, c17_1 ---------------------------------------- (201) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ST(s(s(x0)), cons(s(x1), cons(z2, z3))) -> c1(COND1(or(equal(s(s(x0)), s(x1)), or(equal(s(s(x0)), z2), member(s(s(x0)), z3))), s(s(x0)), cons(s(x1), cons(z2, z3))), MEMBER(s(s(x0)), cons(s(x1), cons(z2, z3)))) by ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c19_2, c17_1 ---------------------------------------- (203) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ST(s(0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(0), 0), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(0, cons(z2, z3))), MEMBER(s(0), cons(0, cons(z2, z3)))) by ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c19_2, c17_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ST(s(0), cons(s(x0), cons(z2, z3))) -> c1(COND1(or(equal(s(0), s(x0)), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(s(x0), cons(z2, z3))), MEMBER(s(0), cons(s(x0), cons(z2, z3)))) by ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c4_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c19_2, c17_1 ---------------------------------------- (207) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND1(false, x0, cons(x1, x2)) -> c4(MAX(cons(x1, x2))) by COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c4_1, c19_2, c17_1 ---------------------------------------- (209) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ST(s(x0), cons(0, cons(z2, z3))) -> c1(COND1(or(equal(s(x0), 0), or(equal(s(x0), z2), member(s(x0), z3))), s(x0), cons(0, cons(z2, z3))), MEMBER(s(x0), cons(0, cons(z2, z3)))) by ST(s(z0), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(z0), z1), member(s(z0), z2))), s(z0), cons(0, cons(z1, z2))), MEMBER(s(z0), cons(0, cons(z1, z2)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) ST(s(z0), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(z0), z1), member(s(z0), z2))), s(z0), cons(0, cons(z1, z2))), MEMBER(s(z0), cons(0, cons(z1, z2)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) ST(s(z0), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(z0), z1), member(s(z0), z2))), s(z0), cons(0, cons(z1, z2))), MEMBER(s(z0), cons(0, cons(z1, z2)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c7_1, c8_1, c20_1, c1_2, c_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c4_1, c19_2, c17_1 ---------------------------------------- (211) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(z0, cons(z1, z2)) -> c7(EQUAL(z0, z1)) by MEMBER(s(s(y0)), cons(s(s(y1)), z2)) -> c7(EQUAL(s(s(y0)), s(s(y1)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: member(z0, nil) -> false member(z0, cons(z1, z2)) -> or(equal(z0, z1), member(z0, z2)) or(z0, true) -> true or(z0, false) -> z0 equal(0, 0) -> true equal(s(z0), 0) -> false equal(0, s(z0)) -> false equal(s(z0), s(z1)) -> equal(z0, z1) gt(0, z0) -> false gt(s(z0), 0) -> true gt(s(z0), s(z1)) -> gt(z0, z1) max(nil) -> 0 max(cons(z0, z1)) -> if(gt(z0, max(z1)), z0, max(z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Tuples: MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) COND1(false, z0, cons(z1, z2)) -> c4(GT(z0, if(gt(z1, max(z2)), z1, max(z2)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) ST(s(z0), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(z0), z1), member(s(z0), z2))), s(z0), cons(0, cons(z1, z2))), MEMBER(s(z0), cons(0, cons(z1, z2)))) MEMBER(s(s(y0)), cons(s(s(y1)), z2)) -> c7(EQUAL(s(s(y0)), s(s(y1)))) S tuples: MEMBER(z0, cons(z1, z2)) -> c8(MEMBER(z0, z2)) MAX(cons(z0, z1)) -> c20(MAX(z1)) ST(s(z0), cons(0, x2)) -> c1(COND1(or(false, member(s(z0), x2)), s(z0), cons(0, x2)), MEMBER(s(z0), cons(0, x2))) MAX(cons(x0, cons(x1, x2))) -> c19(MAX(cons(x1, x2))) COND1(true, x0, cons(x1, cons(x2, x3))) -> c2(ST(s(x0), cons(x1, cons(x2, x3)))) COND1(true, s(x0), cons(0, x1)) -> c2(ST(s(s(x0)), cons(0, x1))) COND1(true, s(x0), cons(s(x1), x2)) -> c2(ST(s(s(x0)), cons(s(x1), x2))) COND1(true, x0, cons(x1, nil)) -> c2(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(0, x1)) -> c5(ST(s(x0), cons(0, x1))) COND2(false, x0, cons(x1, nil)) -> c5(ST(s(x0), cons(x1, nil))) COND2(false, x0, cons(x1, cons(x2, x3))) -> c5(ST(s(x0), cons(x1, cons(x2, x3)))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(COND1(or(equal(s(x0), x1), or(equal(s(x0), x2), member(s(x0), x3))), s(x0), cons(x1, cons(x2, x3))), MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(0), cons(z1, cons(z2, z3))) -> c1(COND1(or(equal(s(0), z1), or(equal(s(0), z2), member(s(0), z3))), s(0), cons(z1, cons(z2, z3))), MEMBER(s(0), cons(z1, cons(z2, z3)))) ST(s(z0), cons(0, nil)) -> c1(COND1(false, s(z0), cons(0, nil)), MEMBER(s(z0), cons(0, nil))) ST(s(z0), cons(s(z1), nil)) -> c1(COND1(equal(z0, z1), s(z0), cons(s(z1), nil)), MEMBER(s(z0), cons(s(z1), nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, max(cons(z2, z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, max(cons(z2, z3)))), MAX(cons(z1, cons(z2, z3)))) ST(s(x0), cons(s(z1), cons(x2, x3))) -> c1(COND1(or(equal(x0, z1), member(s(x0), cons(x2, x3))), s(x0), cons(s(z1), cons(x2, x3))), MEMBER(s(x0), cons(s(z1), cons(x2, x3)))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(COND1(or(equal(s(x0), x1), member(s(s(x0)), x2)), s(s(x0)), cons(s(x1), x2)), MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(s(x0), x1)) -> c1(COND1(or(equal(0, x0), member(s(0), x1)), s(0), cons(s(x0), x1)), MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(s(z1), nil)) -> c1(COND1(or(equal(x0, z1), member(s(x0), nil)), s(x0), cons(s(z1), nil)), MEMBER(s(x0), cons(s(z1), nil))) EQUAL(s(s(y0)), s(s(y1))) -> c14(EQUAL(s(y0), s(y1))) MAX(cons(z0, cons(z1, nil))) -> c19(GT(z0, if(gt(z1, 0), z1, 0)), MAX(cons(z1, nil))) MAX(cons(z0, cons(z1, cons(z2, z3)))) -> c19(GT(z0, if(gt(z1, if(gt(z2, max(z3)), z2, max(z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) MAX(cons(z0, cons(0, z1))) -> c19(GT(z0, max(z1)), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, max(nil)), z1, 0)), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) COND1(false, z0, cons(z1, cons(z2, z3))) -> c3(COND2(gt(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), z0, cons(z1, cons(z2, z3))), GT(z0, if(gt(z1, max(cons(z2, z3))), z1, if(gt(z2, max(z3)), z2, max(z3)))), MAX(cons(z1, cons(z2, z3)))) COND1(false, z0, cons(0, z1)) -> c3(COND2(gt(z0, if(false, 0, max(z1))), z0, cons(0, z1)), GT(z0, if(false, 0, max(z1))), MAX(cons(0, z1))) COND1(false, z0, cons(z1, nil)) -> c3(COND2(gt(z0, if(gt(z1, 0), z1, max(nil))), z0, cons(z1, nil)), GT(z0, if(gt(z1, max(nil)), z1, 0)), MAX(cons(z1, nil))) GT(s(s(y0)), s(s(y1))) -> c17(GT(s(y0), s(y1))) ST(s(s(z0)), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(s(z0)), z1), member(s(s(z0)), z2))), s(s(z0)), cons(0, cons(z1, z2))), MEMBER(s(s(z0)), cons(0, cons(z1, z2)))) ST(s(s(z0)), cons(s(z1), cons(z2, z3))) -> c1(COND1(or(equal(s(z0), z1), or(equal(s(s(z0)), z2), member(s(s(z0)), z3))), s(s(z0)), cons(s(z1), cons(z2, z3))), MEMBER(s(s(z0)), cons(s(z1), cons(z2, z3)))) ST(s(0), cons(0, cons(z0, z1))) -> c1(COND1(or(false, or(equal(s(0), z0), member(s(0), z1))), s(0), cons(0, cons(z0, z1))), MEMBER(s(0), cons(0, cons(z0, z1)))) ST(s(0), cons(s(z0), cons(z1, z2))) -> c1(COND1(or(equal(0, z0), or(equal(s(0), z1), member(s(0), z2))), s(0), cons(s(z0), cons(z1, z2))), MEMBER(s(0), cons(s(z0), cons(z1, z2)))) ST(s(z0), cons(0, cons(z1, z2))) -> c1(COND1(or(false, or(equal(s(z0), z1), member(s(z0), z2))), s(z0), cons(0, cons(z1, z2))), MEMBER(s(z0), cons(0, cons(z1, z2)))) MEMBER(s(s(y0)), cons(s(s(y1)), z2)) -> c7(EQUAL(s(s(y0)), s(s(y1)))) K tuples: ST(0, cons(0, x2)) -> c(COND1(or(true, member(0, x2)), 0, cons(0, x2))) ST(0, cons(s(z0), x2)) -> c(COND1(or(false, member(0, x2)), 0, cons(s(z0), x2))) COND1(true, 0, cons(0, x0)) -> c2(ST(s(0), cons(0, x0))) COND1(true, 0, cons(s(x0), x1)) -> c2(ST(s(0), cons(s(x0), x1))) COND2(false, 0, x0) -> c5(ST(s(0), x0)) COND2(false, 0, cons(x0, x1)) -> c5(ST(s(0), cons(x0, x1))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(COND2(false, 0, cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, cons(x2, x3))) -> c6(MAX(cons(x1, cons(x2, x3)))) COND1(false, 0, cons(x1, x2)) -> c9(COND2(false, 0, cons(x1, x2))) COND1(false, 0, cons(x1, x2)) -> c9(MAX(cons(x1, x2))) COND1(false, 0, cons(0, x0)) -> c9(COND2(false, 0, cons(0, x0))) COND1(false, 0, cons(0, x0)) -> c9(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c9(COND2(false, 0, cons(s(x0), x1))) COND1(false, 0, cons(s(x0), x1)) -> c9(MAX(cons(s(x0), x1))) COND1(false, 0, cons(x1, nil)) -> c9(COND2(false, 0, cons(x1, nil))) COND1(false, 0, cons(x1, nil)) -> c9(MAX(cons(x1, nil))) ST(x0, cons(x1, nil)) -> c1(MEMBER(x0, cons(x1, nil))) ST(s(x0), cons(x1, cons(x2, x3))) -> c1(MEMBER(s(x0), cons(x1, cons(x2, x3)))) ST(s(s(x0)), cons(0, x1)) -> c1(MEMBER(s(s(x0)), cons(0, x1))) ST(s(s(x0)), cons(s(x1), x2)) -> c1(MEMBER(s(s(x0)), cons(s(x1), x2))) ST(s(0), cons(0, x0)) -> c1(MEMBER(s(0), cons(0, x0))) ST(s(0), cons(s(x0), x1)) -> c1(MEMBER(s(0), cons(s(x0), x1))) ST(s(x0), cons(x1, nil)) -> c1(MEMBER(s(x0), cons(x1, nil))) ST(s(0), cons(z1, z2)) -> c1(MEMBER(s(0), cons(z1, z2))) ST(s(x0), cons(0, x1)) -> c1(MEMBER(s(x0), cons(0, x1))) COND1(false, s(x0), cons(0, x1)) -> c4(MAX(cons(0, x1))) COND1(false, 0, cons(0, x0)) -> c4(MAX(cons(0, x0))) COND1(false, 0, cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(x0), cons(x1, cons(x2, x3))) -> c4(MAX(cons(x1, cons(x2, x3)))) COND1(false, s(0), cons(x0, cons(x1, x2))) -> c4(MAX(cons(x0, cons(x1, x2)))) COND1(false, s(x0), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(x0), cons(0, nil)) -> c4(MAX(cons(0, nil))) COND1(false, s(x0), cons(s(x1), nil)) -> c4(MAX(cons(s(x1), nil))) COND1(false, s(x0), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(s(x0)), cons(s(x1), x2)) -> c4(MAX(cons(s(x1), x2))) COND1(false, s(0), cons(s(x0), x1)) -> c4(MAX(cons(s(x0), x1))) COND1(false, s(s(x0)), cons(0, cons(x1, x2))) -> c4(MAX(cons(0, cons(x1, x2)))) COND1(false, s(s(x0)), cons(s(x1), cons(x2, x3))) -> c4(MAX(cons(s(x1), cons(x2, x3)))) COND1(false, s(0), cons(0, cons(x0, x1))) -> c4(MAX(cons(0, cons(x0, x1)))) COND1(false, s(0), cons(s(x0), cons(x1, x2))) -> c4(MAX(cons(s(x0), cons(x1, x2)))) Defined Rule Symbols: member_2, or_2, equal_2, gt_2, max_1, if_3 Defined Pair Symbols: MEMBER_2, MAX_1, ST_2, COND1_3, COND2_3, EQUAL_2, GT_2 Compound Symbols: c8_1, c20_1, c1_2, c_1, c19_1, c2_1, c5_1, c6_1, c1_1, c9_1, c3_3, c14_1, c4_1, c19_2, c17_1, c7_1