KILLED proof of input_ks3QBOB79e.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 70 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 6 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 346 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 171 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 91 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 192 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 34 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 92 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 14 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 159 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 623 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 222 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 100 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 834 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 334 ms] (68) CpxRNTS (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 6968 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 1810 ms] (74) CpxRNTS (75) CompletionProof [UPPER BOUND(ID), 0 ms] (76) CpxTypedWeightedCompleteTrs (77) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 12 ms] (78) CpxRNTS (79) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 98 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 120 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 101 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 110 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 120 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms] (196) CdtProblem (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 182 ms] (230) CdtProblem (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (232) 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: car(cons(x, l)) -> x cddr(nil) -> nil cddr(cons(x, nil)) -> nil cddr(cons(x, cons(y, l))) -> l cadr(cons(x, cons(y, l))) -> y isZero(0) -> true isZero(s(x)) -> false plus(x, y) -> ifplus(isZero(x), x, y) ifplus(true, x, y) -> y ifplus(false, x, y) -> s(plus(p(x), y)) times(x, y) -> iftimes(isZero(x), x, y) iftimes(true, x, y) -> 0 iftimes(false, x, y) -> plus(y, times(p(x), y)) p(s(x)) -> x p(0) -> 0 shorter(nil, y) -> true shorter(cons(x, l), 0) -> false shorter(cons(x, l), s(y)) -> shorter(l, y) prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) if(true, b, l) -> s(0) if(false, b, l) -> if2(b, l) if2(true, l) -> car(l) if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) 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: car(cons(x, l)) -> x cddr(nil) -> nil cddr(cons(x, nil)) -> nil cddr(cons(x, cons(y, l))) -> l cadr(cons(x, cons(y, l))) -> y isZero(0') -> true isZero(s(x)) -> false plus(x, y) -> ifplus(isZero(x), x, y) ifplus(true, x, y) -> y ifplus(false, x, y) -> s(plus(p(x), y)) times(x, y) -> iftimes(isZero(x), x, y) iftimes(true, x, y) -> 0' iftimes(false, x, y) -> plus(y, times(p(x), y)) p(s(x)) -> x p(0') -> 0' shorter(nil, y) -> true shorter(cons(x, l), 0') -> false shorter(cons(x, l), s(y)) -> shorter(l, y) prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l) if(true, b, l) -> s(0') if(false, b, l) -> if2(b, l) if2(true, l) -> car(l) if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) 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: car(cons(x, l)) -> x cddr(nil) -> nil cddr(cons(x, nil)) -> nil cddr(cons(x, cons(y, l))) -> l cadr(cons(x, cons(y, l))) -> y isZero(0) -> true isZero(s(x)) -> false plus(x, y) -> ifplus(isZero(x), x, y) ifplus(true, x, y) -> y ifplus(false, x, y) -> s(plus(p(x), y)) times(x, y) -> iftimes(isZero(x), x, y) iftimes(true, x, y) -> 0 iftimes(false, x, y) -> plus(y, times(p(x), y)) p(s(x)) -> x p(0) -> 0 shorter(nil, y) -> true shorter(cons(x, l), 0) -> false shorter(cons(x, l), s(y)) -> shorter(l, y) prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) if(true, b, l) -> s(0) if(false, b, l) -> if2(b, l) if2(true, l) -> car(l) if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) 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: car(cons(x, l)) -> x [1] cddr(nil) -> nil [1] cddr(cons(x, nil)) -> nil [1] cddr(cons(x, cons(y, l))) -> l [1] cadr(cons(x, cons(y, l))) -> y [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] plus(x, y) -> ifplus(isZero(x), x, y) [1] ifplus(true, x, y) -> y [1] ifplus(false, x, y) -> s(plus(p(x), y)) [1] times(x, y) -> iftimes(isZero(x), x, y) [1] iftimes(true, x, y) -> 0 [1] iftimes(false, x, y) -> plus(y, times(p(x), y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] shorter(nil, y) -> true [1] shorter(cons(x, l), 0) -> false [1] shorter(cons(x, l), s(y)) -> shorter(l, y) [1] prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1] if(true, b, l) -> s(0) [1] if(false, b, l) -> if2(b, l) [1] if2(true, l) -> car(l) [1] if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [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: car(cons(x, l)) -> x [1] cddr(nil) -> nil [1] cddr(cons(x, nil)) -> nil [1] cddr(cons(x, cons(y, l))) -> l [1] cadr(cons(x, cons(y, l))) -> y [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] plus(x, y) -> ifplus(isZero(x), x, y) [1] ifplus(true, x, y) -> y [1] ifplus(false, x, y) -> s(plus(p(x), y)) [1] times(x, y) -> iftimes(isZero(x), x, y) [1] iftimes(true, x, y) -> 0 [1] iftimes(false, x, y) -> plus(y, times(p(x), y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] shorter(nil, y) -> true [1] shorter(cons(x, l), 0) -> false [1] shorter(cons(x, l), s(y)) -> shorter(l, y) [1] prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1] if(true, b, l) -> s(0) [1] if(false, b, l) -> if2(b, l) [1] if2(true, l) -> car(l) [1] if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [1] The TRS has the following type information: car :: cons:nil -> 0:s cons :: 0:s -> cons:nil -> cons:nil cddr :: cons:nil -> cons:nil nil :: cons:nil cadr :: cons:nil -> 0:s isZero :: 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false plus :: 0:s -> 0:s -> 0:s ifplus :: true:false -> 0:s -> 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s iftimes :: true:false -> 0:s -> 0:s -> 0:s shorter :: cons:nil -> 0:s -> true:false prod :: cons:nil -> 0:s if :: true:false -> true:false -> cons:nil -> 0:s if2 :: true:false -> cons:nil -> 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: prod_1 if_3 if2_2 (c) The following functions are completely defined: times_2 car_1 cadr_1 cddr_1 p_1 isZero_1 shorter_2 iftimes_3 plus_2 ifplus_3 Due to the following rules being added: car(v0) -> 0 [0] cadr(v0) -> 0 [0] 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: car(cons(x, l)) -> x [1] cddr(nil) -> nil [1] cddr(cons(x, nil)) -> nil [1] cddr(cons(x, cons(y, l))) -> l [1] cadr(cons(x, cons(y, l))) -> y [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] plus(x, y) -> ifplus(isZero(x), x, y) [1] ifplus(true, x, y) -> y [1] ifplus(false, x, y) -> s(plus(p(x), y)) [1] times(x, y) -> iftimes(isZero(x), x, y) [1] iftimes(true, x, y) -> 0 [1] iftimes(false, x, y) -> plus(y, times(p(x), y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] shorter(nil, y) -> true [1] shorter(cons(x, l), 0) -> false [1] shorter(cons(x, l), s(y)) -> shorter(l, y) [1] prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1] if(true, b, l) -> s(0) [1] if(false, b, l) -> if2(b, l) [1] if2(true, l) -> car(l) [1] if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [1] car(v0) -> 0 [0] cadr(v0) -> 0 [0] The TRS has the following type information: car :: cons:nil -> 0:s cons :: 0:s -> cons:nil -> cons:nil cddr :: cons:nil -> cons:nil nil :: cons:nil cadr :: cons:nil -> 0:s isZero :: 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false plus :: 0:s -> 0:s -> 0:s ifplus :: true:false -> 0:s -> 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s iftimes :: true:false -> 0:s -> 0:s -> 0:s shorter :: cons:nil -> 0:s -> true:false prod :: cons:nil -> 0:s if :: true:false -> true:false -> cons:nil -> 0:s if2 :: true:false -> cons:nil -> 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: car(cons(x, l)) -> x [1] cddr(nil) -> nil [1] cddr(cons(x, nil)) -> nil [1] cddr(cons(x, cons(y, l))) -> l [1] cadr(cons(x, cons(y, l))) -> y [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] plus(0, y) -> ifplus(true, 0, y) [2] plus(s(x'), y) -> ifplus(false, s(x'), y) [2] ifplus(true, x, y) -> y [1] ifplus(false, s(x''), y) -> s(plus(x'', y)) [2] ifplus(false, 0, y) -> s(plus(0, y)) [2] times(0, y) -> iftimes(true, 0, y) [2] times(s(x1), y) -> iftimes(false, s(x1), y) [2] iftimes(true, x, y) -> 0 [1] iftimes(false, s(x2), y) -> plus(y, times(x2, y)) [2] iftimes(false, 0, y) -> plus(y, times(0, y)) [2] p(s(x)) -> x [1] p(0) -> 0 [1] shorter(nil, y) -> true [1] shorter(cons(x, l), 0) -> false [1] shorter(cons(x, l), s(y)) -> shorter(l, y) [1] prod(nil) -> if(true, true, nil) [3] prod(cons(x3, l')) -> if(false, shorter(l', 0), cons(x3, l')) [3] if(true, b, l) -> s(0) [1] if(false, b, l) -> if2(b, l) [1] if2(true, l) -> car(l) [1] if2(false, cons(x4, cons(y', l1))) -> prod(cons(times(x4, y'), l1)) [4] if2(false, cons(x4, nil)) -> prod(cons(times(x4, 0), nil)) [3] if2(false, cons(x4, cons(y1, l3))) -> prod(cons(times(x4, 0), l3)) [3] if2(false, cons(x5, cons(y'', l2))) -> prod(cons(times(0, y''), l2)) [3] if2(false, nil) -> prod(cons(times(0, 0), nil)) [2] if2(false, cons(x6, nil)) -> prod(cons(times(0, 0), nil)) [2] if2(false, cons(x7, cons(y2, l4))) -> prod(cons(times(0, 0), l4)) [2] car(v0) -> 0 [0] cadr(v0) -> 0 [0] The TRS has the following type information: car :: cons:nil -> 0:s cons :: 0:s -> cons:nil -> cons:nil cddr :: cons:nil -> cons:nil nil :: cons:nil cadr :: cons:nil -> 0:s isZero :: 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false plus :: 0:s -> 0:s -> 0:s ifplus :: true:false -> 0:s -> 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s iftimes :: true:false -> 0:s -> 0:s -> 0:s shorter :: cons:nil -> 0:s -> true:false prod :: cons:nil -> 0:s if :: true:false -> true:false -> cons:nil -> 0:s if2 :: true:false -> cons:nil -> 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: nil => 0 0 => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0 if(z, z', z'') -{ 1 }-> if2(b, l) :|: b >= 0, z' = b, l >= 0, z = 0, z'' = l if(z, z', z'') -{ 1 }-> 1 + 0 :|: b >= 0, z = 1, z' = b, l >= 0, z'' = l if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + 0) :|: x4 >= 0, z = 0, z' = 1 + x4 + 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' = 1 + x6 + 0, x6 >= 0, z = 0 if2(z, z') -{ 1 }-> car(l) :|: z' = l, z = 1, l >= 0 ifplus(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(x'', y) :|: z' = 1 + x'', z'' = y, y >= 0, x'' >= 0, z = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(y, times(x2, y)) :|: z' = 1 + x2, z'' = y, y >= 0, z = 0, x2 >= 0 iftimes(z, z', z'') -{ 2 }-> plus(y, times(0, y)) :|: z'' = y, y >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 plus(z, z') -{ 2 }-> ifplus(1, 0, y) :|: y >= 0, z = 0, z' = y plus(z, z') -{ 2 }-> ifplus(0, 1 + x', y) :|: z = 1 + x', x' >= 0, y >= 0, z' = y prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, y) :|: z' = 1 + y, x >= 0, y >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, y) :|: y >= 0, z = 0, z' = y times(z, z') -{ 2 }-> iftimes(0, 1 + x1, y) :|: x1 >= 0, y >= 0, z = 1 + x1, z' = y ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0 if(z, z', z'') -{ 1 }-> if2(b, l) :|: b >= 0, z' = b, l >= 0, z = 0, z'' = l if(z, z', z'') -{ 1 }-> 1 + 0 :|: b >= 0, z = 1, z' = b, l >= 0, z'' = l if2(z, z') -{ 2 }-> x :|: z' = l, z = 1, l >= 0, x >= 0, l' >= 0, l = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + 0) :|: x4 >= 0, z = 0, z' = 1 + x4 + 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' = 1 + x6 + 0, x6 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z' = l, z = 1, l >= 0, v0 >= 0, l = v0 ifplus(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(x'', y) :|: z' = 1 + x'', z'' = y, y >= 0, x'' >= 0, z = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(y, times(x2, y)) :|: z' = 1 + x2, z'' = y, y >= 0, z = 0, x2 >= 0 iftimes(z, z', z'') -{ 2 }-> plus(y, times(0, y)) :|: z'' = y, y >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 plus(z, z') -{ 2 }-> ifplus(1, 0, y) :|: y >= 0, z = 0, z' = y plus(z, z') -{ 2 }-> ifplus(0, 1 + x', y) :|: z = 1 + x', x' >= 0, y >= 0, z' = y prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, y) :|: z' = 1 + y, x >= 0, y >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, y) :|: y >= 0, z = 0, z' = y times(z, z') -{ 2 }-> iftimes(0, 1 + x1, y) :|: x1 >= 0, y >= 0, z = 1 + x1, z' = y ---------------------------------------- (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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { shorter } { isZero } { cddr } { car } { cadr } { plus, ifplus } { p } { iftimes, times } { if2, if, prod } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {shorter}, {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} ---------------------------------------- (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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {shorter}, {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: shorter 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {shorter}, {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: shorter after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: isZero 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: isZero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: cddr after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: ?, size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cddr 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: car after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: ?, size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: car 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: cadr 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: ?, size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cadr after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], 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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' Computed SIZE bound using CoFloCo for: ifplus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ifplus: runtime: ?, size: O(n^1) [1 + z' + z''] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 4*z Computed RUNTIME bound using CoFloCo for: ifplus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 8 + 4*z' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0 ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] ---------------------------------------- (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: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {p}, {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (63) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: iftimes after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z'*z'' + 2*z'' Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z*z' + 4*z' ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {iftimes,times}, {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] iftimes: runtime: ?, size: O(n^2) [z'*z'' + 2*z''] times: runtime: ?, size: O(n^2) [z*z' + 4*z'] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: iftimes after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 21 + 10*z' + 4*z'*z'' + 8*z'' Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 46 + 10*z + 4*z*z' + 16*z' ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0 if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0 times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z''] times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z'] ---------------------------------------- (69) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 50 + 10*x4 + 4*x4*y' + 16*y' }-> prod(1 + s10 + l1) :|: s10 >= 0, s10 <= y' * x4 + 4 * y', z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 39 + 10*z' }-> prod(1 + s11 + 0) :|: s11 >= 0, s11 <= 0 * (z' - 1) + 4 * 0, z' - 1 >= 0, z = 0 if2(z, z') -{ 49 + 10*x4 }-> prod(1 + s12 + l3) :|: s12 >= 0, s12 <= 0 * x4 + 4 * 0, y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 49 + 16*y'' }-> prod(1 + s13 + l2) :|: s13 >= 0, s13 <= y'' * 0 + 4 * y'', x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 48 }-> prod(1 + s14 + 0) :|: s14 >= 0, s14 <= 0 * 0 + 4 * 0, z = 0, z' = 0 if2(z, z') -{ 48 }-> prod(1 + s15 + 0) :|: s15 >= 0, s15 <= 0 * 0 + 4 * 0, z' - 1 >= 0, z = 0 if2(z, z') -{ 48 }-> prod(1 + s16 + l4) :|: s16 >= 0, s16 <= 0 * 0 + 4 * 0, x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 44 + 10*z' + 4*z'*z'' + 16*z'' }-> s7 :|: s6 >= 0, s6 <= z'' * (z' - 1) + 4 * z'', s7 >= 0, s7 <= z'' + s6, z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 54 + 20*z'' }-> s9 :|: s8 >= 0, s8 <= z'' * 0 + 4 * z'', s9 >= 0, s9 <= z'' + s8, z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 23 + 8*z' }-> s4 :|: s4 >= 0, s4 <= 0 * z' + 2 * z', z' >= 0, z = 0 times(z, z') -{ 23 + 10*z + 4*z*z' + 8*z' }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 1)) * z' + 2 * z', z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z''] times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z'] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: prod after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 50 + 10*x4 + 4*x4*y' + 16*y' }-> prod(1 + s10 + l1) :|: s10 >= 0, s10 <= y' * x4 + 4 * y', z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 39 + 10*z' }-> prod(1 + s11 + 0) :|: s11 >= 0, s11 <= 0 * (z' - 1) + 4 * 0, z' - 1 >= 0, z = 0 if2(z, z') -{ 49 + 10*x4 }-> prod(1 + s12 + l3) :|: s12 >= 0, s12 <= 0 * x4 + 4 * 0, y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 49 + 16*y'' }-> prod(1 + s13 + l2) :|: s13 >= 0, s13 <= y'' * 0 + 4 * y'', x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 48 }-> prod(1 + s14 + 0) :|: s14 >= 0, s14 <= 0 * 0 + 4 * 0, z = 0, z' = 0 if2(z, z') -{ 48 }-> prod(1 + s15 + 0) :|: s15 >= 0, s15 <= 0 * 0 + 4 * 0, z' - 1 >= 0, z = 0 if2(z, z') -{ 48 }-> prod(1 + s16 + l4) :|: s16 >= 0, s16 <= 0 * 0 + 4 * 0, x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 44 + 10*z' + 4*z'*z'' + 16*z'' }-> s7 :|: s6 >= 0, s6 <= z'' * (z' - 1) + 4 * z'', s7 >= 0, s7 <= z'' + s6, z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 54 + 20*z'' }-> s9 :|: s8 >= 0, s8 <= z'' * 0 + 4 * z'', s9 >= 0, s9 <= z'' + s8, z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 23 + 8*z' }-> s4 :|: s4 >= 0, s4 <= 0 * z' + 2 * z', z' >= 0, z = 0 times(z, z') -{ 23 + 10*z + 4*z*z' + 8*z' }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 1)) * z' + 2 * z', z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z''] times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z'] if2: runtime: ?, size: INF if: runtime: ?, size: INF prod: runtime: ?, size: INF ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: z >= 0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: z >= 0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0 if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l' if2(z, z') -{ 50 + 10*x4 + 4*x4*y' + 16*y' }-> prod(1 + s10 + l1) :|: s10 >= 0, s10 <= y' * x4 + 4 * y', z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0 if2(z, z') -{ 39 + 10*z' }-> prod(1 + s11 + 0) :|: s11 >= 0, s11 <= 0 * (z' - 1) + 4 * 0, z' - 1 >= 0, z = 0 if2(z, z') -{ 49 + 10*x4 }-> prod(1 + s12 + l3) :|: s12 >= 0, s12 <= 0 * x4 + 4 * 0, y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0 if2(z, z') -{ 49 + 16*y'' }-> prod(1 + s13 + l2) :|: s13 >= 0, s13 <= y'' * 0 + 4 * y'', x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0 if2(z, z') -{ 48 }-> prod(1 + s14 + 0) :|: s14 >= 0, s14 <= 0 * 0 + 4 * 0, z = 0, z' = 0 if2(z, z') -{ 48 }-> prod(1 + s15 + 0) :|: s15 >= 0, s15 <= 0 * 0 + 4 * 0, z' - 1 >= 0, z = 0 if2(z, z') -{ 48 }-> prod(1 + s16 + l4) :|: s16 >= 0, s16 <= 0 * 0 + 4 * 0, x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0 if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0 ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0 ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 44 + 10*z' + 4*z'*z'' + 16*z'' }-> s7 :|: s6 >= 0, s6 <= z'' * (z' - 1) + 4 * z'', s7 >= 0, s7 <= z'' + s6, z'' >= 0, z = 0, z' - 1 >= 0 iftimes(z, z', z'') -{ 54 + 20*z'' }-> s9 :|: s8 >= 0, s8 <= z'' * 0 + 4 * z'', s9 >= 0, s9 <= z'' + s8, z'' >= 0, z = 0, z' = 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 isZero(z) -{ 1 }-> 1 :|: z = 0 isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0 plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0 prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0 prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0 shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 times(z, z') -{ 23 + 8*z' }-> s4 :|: s4 >= 0, s4 <= 0 * z' + 2 * z', z' >= 0, z = 0 times(z, z') -{ 23 + 10*z + 4*z*z' + 8*z' }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 1)) * z' + 2 * z', z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if2,if,prod} Previous analysis results are: shorter: runtime: O(n^1) [1 + z'], size: O(1) [1] isZero: runtime: O(1) [1], size: O(1) [1] cddr: runtime: O(1) [1], size: O(n^1) [z] car: runtime: O(1) [1], size: O(n^1) [z] cadr: runtime: O(1) [1], size: O(n^1) [z] plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z'] ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z''] times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z'] if2: runtime: INF, size: INF if: runtime: ?, size: INF prod: runtime: ?, size: INF ---------------------------------------- (75) 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: car(v0) -> null_car [0] cadr(v0) -> null_cadr [0] isZero(v0) -> null_isZero [0] p(v0) -> null_p [0] shorter(v0, v1) -> null_shorter [0] ifplus(v0, v1, v2) -> null_ifplus [0] iftimes(v0, v1, v2) -> null_iftimes [0] if(v0, v1, v2) -> null_if [0] if2(v0, v1) -> null_if2 [0] And the following fresh constants: null_car, null_cadr, null_isZero, null_p, null_shorter, null_ifplus, null_iftimes, null_if, null_if2 ---------------------------------------- (76) 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: car(cons(x, l)) -> x [1] cddr(nil) -> nil [1] cddr(cons(x, nil)) -> nil [1] cddr(cons(x, cons(y, l))) -> l [1] cadr(cons(x, cons(y, l))) -> y [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] plus(x, y) -> ifplus(isZero(x), x, y) [1] ifplus(true, x, y) -> y [1] ifplus(false, x, y) -> s(plus(p(x), y)) [1] times(x, y) -> iftimes(isZero(x), x, y) [1] iftimes(true, x, y) -> 0 [1] iftimes(false, x, y) -> plus(y, times(p(x), y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] shorter(nil, y) -> true [1] shorter(cons(x, l), 0) -> false [1] shorter(cons(x, l), s(y)) -> shorter(l, y) [1] prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1] if(true, b, l) -> s(0) [1] if(false, b, l) -> if2(b, l) [1] if2(true, l) -> car(l) [1] if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [1] car(v0) -> null_car [0] cadr(v0) -> null_cadr [0] isZero(v0) -> null_isZero [0] p(v0) -> null_p [0] shorter(v0, v1) -> null_shorter [0] ifplus(v0, v1, v2) -> null_ifplus [0] iftimes(v0, v1, v2) -> null_iftimes [0] if(v0, v1, v2) -> null_if [0] if2(v0, v1) -> null_if2 [0] The TRS has the following type information: car :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 cons :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> cons:nil -> cons:nil cddr :: cons:nil -> cons:nil nil :: cons:nil cadr :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 isZero :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> true:false:null_isZero:null_shorter 0 :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 true :: true:false:null_isZero:null_shorter s :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 false :: true:false:null_isZero:null_shorter plus :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 ifplus :: true:false:null_isZero:null_shorter -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 p :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 times :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 iftimes :: true:false:null_isZero:null_shorter -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 shorter :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> true:false:null_isZero:null_shorter prod :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 if :: true:false:null_isZero:null_shorter -> true:false:null_isZero:null_shorter -> cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 if2 :: true:false:null_isZero:null_shorter -> cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_car :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_cadr :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_isZero :: true:false:null_isZero:null_shorter null_p :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_shorter :: true:false:null_isZero:null_shorter null_ifplus :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_iftimes :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_if :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 null_if2 :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 Rewrite Strategy: INNERMOST ---------------------------------------- (77) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 true => 2 false => 1 null_car => 0 null_cadr => 0 null_isZero => 0 null_p => 0 null_shorter => 0 null_ifplus => 0 null_iftimes => 0 null_if => 0 null_if2 => 0 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cadr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l) cddr(z) -{ 1 }-> 0 :|: z = 0 cddr(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0 if(z, z', z'') -{ 1 }-> if2(b, l) :|: b >= 0, z = 1, z' = b, l >= 0, z'' = l if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 2, b >= 0, z' = b, l >= 0, z'' = l if2(z, z') -{ 1 }-> prod(1 + times(car(l), cadr(l)) + cddr(l)) :|: z' = l, z = 1, l >= 0 if2(z, z') -{ 1 }-> car(l) :|: z = 2, z' = l, l >= 0 if2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ifplus(z, z', z'') -{ 1 }-> y :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 ifplus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ifplus(z, z', z'') -{ 1 }-> 1 + plus(p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 iftimes(z, z', z'') -{ 1 }-> plus(y, times(p(x), y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 iftimes(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 isZero(z) -{ 1 }-> 2 :|: z = 0 isZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x isZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 plus(z, z') -{ 1 }-> ifplus(isZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y prod(z) -{ 1 }-> if(shorter(l, 0), shorter(l, 1 + 0), l) :|: z = l, l >= 0 shorter(z, z') -{ 1 }-> shorter(l, y) :|: z' = 1 + y, x >= 0, y >= 0, l >= 0, z = 1 + x + l shorter(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y shorter(z, z') -{ 1 }-> 1 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0 shorter(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 times(z, z') -{ 1 }-> iftimes(isZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (79) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: car(cons(z0, z1)) -> z0 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 cadr(cons(z0, cons(z1, z2))) -> z1 isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) p(s(z0)) -> z0 p(0) -> 0 shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0) if(true, z0, z1) -> s(0) if(false, z0, z1) -> if2(z0, z1) if2(true, z0) -> car(z0) if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0))) Tuples: CAR(cons(z0, z1)) -> c CDDR(nil) -> c1 CDDR(cons(z0, nil)) -> c2 CDDR(cons(z0, cons(z1, z2))) -> c3 CADR(cons(z0, cons(z1, z2))) -> c4 ISZERO(0) -> c5 ISZERO(s(z0)) -> c6 PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0)) IFPLUS(true, z0, z1) -> c8 IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0)) IFTIMES(true, z0, z1) -> c11 IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0)) P(s(z0)) -> c13 P(0) -> c14 SHORTER(nil, z0) -> c15 SHORTER(cons(z0, z1), 0) -> c16 SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(true, z0, z1) -> c20 IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(true, z0) -> c22(CAR(z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0)) S tuples: CAR(cons(z0, z1)) -> c CDDR(nil) -> c1 CDDR(cons(z0, nil)) -> c2 CDDR(cons(z0, cons(z1, z2))) -> c3 CADR(cons(z0, cons(z1, z2))) -> c4 ISZERO(0) -> c5 ISZERO(s(z0)) -> c6 PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0)) IFPLUS(true, z0, z1) -> c8 IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0)) IFTIMES(true, z0, z1) -> c11 IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0)) P(s(z0)) -> c13 P(0) -> c14 SHORTER(nil, z0) -> c15 SHORTER(cons(z0, z1), 0) -> c16 SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(true, z0, z1) -> c20 IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(true, z0) -> c22(CAR(z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0)) K tuples:none Defined Rule Symbols: car_1, cddr_1, cadr_1, isZero_1, plus_2, ifplus_3, times_2, iftimes_3, p_1, shorter_2, prod_1, if_3, if2_2 Defined Pair Symbols: CAR_1, CDDR_1, CADR_1, ISZERO_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, P_1, SHORTER_2, PROD_1, IF_3, IF2_2 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7_2, c8, c9_2, c10_2, c11, c12_3, c13, c14, c15, c16, c17_1, c18_2, c19_2, c20, c21_1, c22_1, c23_3, c24_3, c25_2 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 15 trailing nodes: IF(true, z0, z1) -> c20 IFTIMES(true, z0, z1) -> c11 CADR(cons(z0, cons(z1, z2))) -> c4 SHORTER(nil, z0) -> c15 ISZERO(0) -> c5 CDDR(cons(z0, cons(z1, z2))) -> c3 IF2(true, z0) -> c22(CAR(z0)) P(s(z0)) -> c13 CDDR(nil) -> c1 CDDR(cons(z0, nil)) -> c2 SHORTER(cons(z0, z1), 0) -> c16 IFPLUS(true, z0, z1) -> c8 ISZERO(s(z0)) -> c6 CAR(cons(z0, z1)) -> c P(0) -> c14 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: car(cons(z0, z1)) -> z0 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 cadr(cons(z0, cons(z1, z2))) -> z1 isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) p(s(z0)) -> z0 p(0) -> 0 shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0) if(true, z0, z1) -> s(0) if(false, z0, z1) -> if2(z0, z1) if2(true, z0) -> car(z0) if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0))) Tuples: PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0)) SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0)) S tuples: PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0)) SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0)) K tuples:none Defined Rule Symbols: car_1, cddr_1, cadr_1, isZero_1, plus_2, ifplus_3, times_2, iftimes_3, p_1, shorter_2, prod_1, if_3, if2_2 Defined Pair Symbols: PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, SHORTER_2, PROD_1, IF_3, IF2_2 Compound Symbols: c7_2, c9_2, c10_2, c12_3, c17_1, c18_2, c19_2, c21_1, c23_3, c24_3, c25_2 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: car(cons(z0, z1)) -> z0 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 cadr(cons(z0, cons(z1, z2))) -> z1 isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) p(s(z0)) -> z0 p(0) -> 0 shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0) if(true, z0, z1) -> s(0) if(false, z0, z1) -> if2(z0, z1) if2(true, z0) -> car(z0) if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0))) Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) K tuples:none Defined Rule Symbols: car_1, cddr_1, cadr_1, isZero_1, plus_2, ifplus_3, times_2, iftimes_3, p_1, shorter_2, prod_1, if_3, if2_2 Defined Pair Symbols: SHORTER_2, PROD_1, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c19_2, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1 ---------------------------------------- (85) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0) if(true, z0, z1) -> s(0) if(false, z0, z1) -> if2(z0, z1) if2(true, z0) -> car(z0) if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) K tuples:none Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, PROD_1, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c19_2, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) by PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples:none Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples:none Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) K tuples:none Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1 ---------------------------------------- (93) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) by PLUS(0, x1) -> c7(IFPLUS(true, 0, x1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(0, x1) -> c7(IFPLUS(true, 0, x1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(0, x1) -> c7(IFPLUS(true, 0, x1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2 Compound Symbols: c17_1, c21_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PLUS(0, x1) -> c7(IFPLUS(true, 0, x1)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2 Compound Symbols: c17_1, c21_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) by IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) IFPLUS(false, 0, x1) -> c9(PLUS(0, x1)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) IFPLUS(false, 0, x1) -> c9(PLUS(0, x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) IFPLUS(false, 0, x1) -> c9(PLUS(0, x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3 Compound Symbols: c17_1, c21_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IFPLUS(false, 0, x1) -> c9(PLUS(0, x1)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3 Compound Symbols: c17_1, c21_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) by TIMES(0, x1) -> c10(IFTIMES(true, 0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(0, x1) -> c10(IFTIMES(true, 0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(0, x1) -> c10(IFTIMES(true, 0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2 Compound Symbols: c17_1, c21_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1 ---------------------------------------- (105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: TIMES(0, x1) -> c10(IFTIMES(true, 0, x1)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2 Compound Symbols: c17_1, c21_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) by IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(0, x1)), TIMES(p(0), x1)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(0, x1)), TIMES(p(0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(0, x1)), TIMES(p(0), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2 ---------------------------------------- (109) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1 ---------------------------------------- (111) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) by PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1 ---------------------------------------- (117) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) by IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_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. IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) We considered the (Usable) Rules:none And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3)) = x_3 POL(IF2(x_1, x_2)) = x_2 POL(IFPLUS(x_1, x_2, x_3)) = x_1 + x_2 POL(IFTIMES(x_1, x_2, x_3)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = x_1 POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = [1] POL(isZero(x_1)) = 0 POL(nil) = [1] POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = 0 POL(times(x_1, x_2)) = 0 POL(true) = 0 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) by IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) We considered the (Usable) Rules:none And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3)) = x_3 POL(IF2(x_1, x_2)) = x_2 POL(IFPLUS(x_1, x_2, x_3)) = x_1 + x_2 POL(IFTIMES(x_1, x_2, x_3)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = x_1 POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = [1] POL(isZero(x_1)) = 0 POL(nil) = [1] POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = 0 POL(times(x_1, x_2)) = 0 POL(true) = 0 ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) by IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1 ---------------------------------------- (133) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) We considered the (Usable) Rules:none And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3)) = x_3 POL(IF2(x_1, x_2)) = x_2 POL(IFPLUS(x_1, x_2, x_3)) = x_1 + x_2 POL(IFTIMES(x_1, x_2, x_3)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = x_1 POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] POL(cons(x_1, x_2)) = 0 POL(false) = 0 POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = [1] POL(isZero(x_1)) = 0 POL(nil) = [1] POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = 0 POL(times(x_1, x_2)) = 0 POL(true) = 0 ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) by PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1 ---------------------------------------- (137) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) We considered the (Usable) Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) shorter(cons(z0, z1), 0) -> false And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(IF(x_1, x_2, x_3)) = x_2 POL(IF2(x_1, x_2)) = x_1 POL(IFPLUS(x_1, x_2, x_3)) = 0 POL(IFTIMES(x_1, x_2, x_3)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = [1] POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] + x_1 POL(cons(x_1, x_2)) = 0 POL(false) = [1] POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = [1] POL(isZero(x_1)) = 0 POL(nil) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = x_1 POL(shorter(x_1, x_2)) = x_2 POL(times(x_1, x_2)) = 0 POL(true) = 0 ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) by PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2 Compound Symbols: c17_1, c21_1, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) by PROD(nil) -> c19(IF(true, true, nil)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) PROD(nil) -> c19(IF(true, true, nil)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples: PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c19_1 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PROD(nil) -> c19(IF(true, true, nil)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) by IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)), TIMES(0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)), TIMES(0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)), TIMES(0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_1 ---------------------------------------- (147) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_1 ---------------------------------------- (149) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_1 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) by IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) by IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) by IFTIMES(false, 0, x0) -> c(TIMES(0, x0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) IFTIMES(false, 0, x0) -> c(TIMES(0, x0)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) K tuples: IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IFTIMES(false, 0, x0) -> c(TIMES(0, x0)) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) K tuples: PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) by PROD(nil) -> c18(IF(true, true, nil)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(nil) -> c18(IF(true, true, nil)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) K tuples: PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PROD(nil) -> c18(IF(true, true, nil)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) by PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1 ---------------------------------------- (165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) We considered the (Usable) Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) shorter(cons(z0, z1), 0) -> false And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3)) = x_1 + x_2 POL(IF2(x_1, x_2)) = [1] + x_1 POL(IFPLUS(x_1, x_2, x_3)) = x_2 POL(IFTIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = [1] + x_1 POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] POL(cons(x_1, x_2)) = [1] POL(false) = [1] POL(ifplus(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = 0 POL(isZero(x_1)) = 0 POL(nil) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = [1] POL(times(x_1, x_2)) = [1] POL(true) = 0 ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) by PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) isZero(0) -> true isZero(s(z0)) -> false p(s(z0)) -> z0 p(0) -> 0 times(z0, z1) -> iftimes(isZero(z0), z0, z1) iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) plus(z0, z1) -> ifplus(isZero(z0), z0, z1) ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) car(cons(z0, z1)) -> z0 cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1 ---------------------------------------- (169) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) by IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) by IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) by IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(z0, z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2 ---------------------------------------- (181) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) We considered the (Usable) Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3)) = x_2 POL(IF2(x_1, x_2)) = x_1 POL(IFPLUS(x_1, x_2, x_3)) = x_2 POL(IFTIMES(x_1, x_2, x_3)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = [1] POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] + x_1 POL(cons(x_1, x_2)) = 0 POL(false) = [1] POL(ifplus(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = 0 POL(isZero(x_1)) = 0 POL(nil) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = [1] + x_2 POL(times(x_1, x_2)) = [1] POL(true) = 0 ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) by IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) by IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) by IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1 ---------------------------------------- (193) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) by IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(z0, z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2 ---------------------------------------- (195) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) We considered the (Usable) Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3)) = x_1 + x_2 POL(IF2(x_1, x_2)) = [1] + x_1 POL(IFPLUS(x_1, x_2, x_3)) = x_2 POL(IFTIMES(x_1, x_2, x_3)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = [1] + x_1 POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] POL(cons(x_1, x_2)) = [1] POL(false) = [1] POL(ifplus(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = 0 POL(isZero(x_1)) = 0 POL(nil) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = [1] + x_2 POL(times(x_1, x_2)) = [1] POL(true) = 0 ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2 ---------------------------------------- (197) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) by IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) by IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) by IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) by IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (205) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) by IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) by IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(z1, z2)))), z2))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(z0, z1), cddr(cons(z0, cons(z1, z2)))))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (209) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) by PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) by PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (213) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) by IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (215) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) by IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) by IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) by IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) by PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1 ---------------------------------------- (223) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) by PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) by IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1 ---------------------------------------- (227) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) by IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1 ---------------------------------------- (229) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) We considered the (Usable) Rules: shorter(nil, z0) -> true shorter(cons(z0, z1), 0) -> false And the Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(IF(x_1, x_2, x_3)) = [1] + x_2 POL(IF2(x_1, x_2)) = [1] + x_1 POL(IFPLUS(x_1, x_2, x_3)) = x_2 POL(IFTIMES(x_1, x_2, x_3)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PROD(x_1)) = [1] + x_1 POL(SHORTER(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c23(x_1, x_2)) = x_1 + x_2 POL(c24(x_1)) = x_1 POL(c24(x_1, x_2)) = x_1 + x_2 POL(c25(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cadr(x_1)) = [1] POL(car(x_1)) = 0 POL(cddr(x_1)) = [1] POL(cons(x_1, x_2)) = [1] POL(false) = [1] POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(iftimes(x_1, x_2, x_3)) = 0 POL(isZero(x_1)) = 0 POL(nil) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(shorter(x_1, x_2)) = x_2 POL(times(x_1, x_2)) = [1] POL(true) = 0 ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1 ---------------------------------------- (231) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) by IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(x1, x2)))), z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> iftimes(isZero(z0), z0, z1) car(cons(z0, z1)) -> z0 iftimes(true, z0, z1) -> 0 iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1)) isZero(0) -> true isZero(s(z0)) -> false plus(z0, z1) -> ifplus(isZero(z0), z0, z1) p(s(z0)) -> z0 p(0) -> 0 ifplus(true, z0, z1) -> z1 ifplus(false, z0, z1) -> s(plus(p(z0), z1)) cadr(cons(z0, cons(z1, z2))) -> z1 cddr(nil) -> nil cddr(cons(z0, nil)) -> nil cddr(cons(z0, cons(z1, z2))) -> z2 shorter(cons(z0, z1), 0) -> false shorter(nil, z0) -> true Tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0))) IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0))) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(x1, x2)))), z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) S tuples: SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2)) IF(false, z0, z1) -> c21(IF2(z0, z1)) PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1)) IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1)) TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1)) IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1))) PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1)) IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1)))) IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0)))) IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1)))) IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil))) IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil))) IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2))) IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2))) IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2)))))) IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2)))))) IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1))))) PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0))) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1)) IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1)) PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(x1, x2)))), z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2))))) K tuples: IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1)))) IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) Defined Rule Symbols: times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2 Defined Pair Symbols: SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3 Compound Symbols: c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1