KILLED proof of input_ZaAo4NJkCr.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) InliningProof [UPPER BOUND(ID), 116 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 134 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 304 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 36 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 69 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 454 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 144 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 437 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 146 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 4638 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 499 ms] (60) CpxRNTS (61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 93 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (150) CdtProblem (151) CdtLeafRemovalProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 17 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 49 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 72 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 26 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms] (208) CdtProblem (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms] (210) CdtProblem (211) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (240) 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: plus(0, x) -> x plus(s(x), y) -> s(plus(x, y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(x, y)) p(s(x)) -> x p(0) -> 0 minus(x, 0) -> x minus(0, x) -> 0 minus(x, s(y)) -> p(minus(x, y)) isZero(0) -> true isZero(s(x)) -> false facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x)) if(true, x, y, z) -> y if(false, x, y, z) -> facIter(x, z) factorial(x) -> facIter(x, s(0)) 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: plus(0', x) -> x plus(s(x), y) -> s(plus(x, y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(x, y)) p(s(x)) -> x p(0') -> 0' minus(x, 0') -> x minus(0', x) -> 0' minus(x, s(y)) -> p(minus(x, y)) isZero(0') -> true isZero(s(x)) -> false facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x)) if(true, x, y, z) -> y if(false, x, y, z) -> facIter(x, z) factorial(x) -> facIter(x, s(0')) 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: plus(0, x) -> x plus(s(x), y) -> s(plus(x, y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(x, y)) p(s(x)) -> x p(0) -> 0 minus(x, 0) -> x minus(0, x) -> 0 minus(x, s(y)) -> p(minus(x, y)) isZero(0) -> true isZero(s(x)) -> false facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x)) if(true, x, y, z) -> y if(false, x, y, z) -> facIter(x, z) factorial(x) -> facIter(x, s(0)) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] minus(x, 0) -> x [1] minus(0, x) -> 0 [1] minus(x, s(y)) -> p(minus(x, y)) [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x)) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> facIter(x, z) [1] factorial(x) -> facIter(x, s(0)) [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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] minus(x, 0) -> x [1] minus(0, x) -> 0 [1] minus(x, s(y)) -> p(minus(x, y)) [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x)) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> facIter(x, z) [1] factorial(x) -> facIter(x, s(0)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s isZero :: 0:s -> true:false true :: true:false false :: true:false facIter :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s factorial :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (10) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] minus(x, 0) -> x [1] minus(0, x) -> 0 [1] minus(x, s(y)) -> p(minus(x, y)) [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x)) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> facIter(x, z) [1] factorial(x) -> facIter(x, s(0)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s isZero :: 0:s -> true:false true :: true:false false :: true:false facIter :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s factorial :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 1 }-> if(isZero(x), minus(x, 1 + 0), y, times(y, x)) :|: z' = x, z'' = y, x >= 0, y >= 0 factorial(z') -{ 1 }-> facIter(x, 1 + 0) :|: z' = x, x >= 0 if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(x, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0 minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 minus(z', z'') -{ 1 }-> p(minus(x, y)) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y minus(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0 p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> plus(y, times(x, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) 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: facIter_2 if_4 factorial_1 (c) The following functions are completely defined: times_2 isZero_1 minus_2 plus_2 p_1 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (14) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] p(s(x)) -> x [1] p(0) -> 0 [1] minus(x, 0) -> x [1] minus(0, x) -> 0 [1] minus(x, s(y)) -> p(minus(x, y)) [1] isZero(0) -> true [1] isZero(s(x)) -> false [1] facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x)) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> facIter(x, z) [1] factorial(x) -> facIter(x, s(0)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s isZero :: 0:s -> true:false true :: true:false false :: true:false facIter :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s factorial :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(0), y) -> plus(y, 0) [2] times(s(s(x')), y) -> plus(y, plus(y, times(x', y))) [2] p(s(x)) -> x [1] p(0) -> 0 [1] minus(x, 0) -> x [1] minus(0, x) -> 0 [1] minus(x, s(0)) -> p(x) [2] minus(0, s(y)) -> p(0) [2] minus(x, s(s(y'))) -> p(p(minus(x, y'))) [2] isZero(0) -> true [1] isZero(s(x)) -> false [1] facIter(0, 0) -> if(true, 0, 0, 0) [4] facIter(0, s(x1)) -> if(true, 0, s(x1), plus(0, times(x1, 0))) [4] facIter(0, 0) -> if(true, p(minus(0, 0)), 0, 0) [4] facIter(0, s(x2)) -> if(true, p(minus(0, 0)), s(x2), plus(0, times(x2, 0))) [4] facIter(s(x''), 0) -> if(false, p(minus(s(x''), 0)), 0, 0) [4] facIter(s(x''), s(x3)) -> if(false, p(minus(s(x''), 0)), s(x3), plus(s(x''), times(x3, s(x'')))) [4] if(true, x, y, z) -> y [1] if(false, x, y, z) -> facIter(x, z) [1] factorial(x) -> facIter(x, s(0)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s isZero :: 0:s -> true:false true :: true:false false :: true:false facIter :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s factorial :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + x2, plus(0, times(x2, 0))) :|: z'' = 1 + x2, x2 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + x1, plus(0, times(x1, 0))) :|: x1 >= 0, z'' = 1 + x1, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + x'', 0)), 0, 0) :|: z'' = 0, z' = 1 + x'', x'' >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + x'', 0)), 1 + x3, plus(1 + x'', times(x3, 1 + x''))) :|: z' = 1 + x'', z'' = 1 + x3, x'' >= 0, x3 >= 0 factorial(z') -{ 1 }-> facIter(x, 1 + 0) :|: z' = x, x >= 0 if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(x, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0 minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 minus(z', z'') -{ 2 }-> p(x) :|: z' = x, x >= 0, z'' = 1 + 0 minus(z', z'') -{ 2 }-> p(p(minus(x, y'))) :|: z' = x, x >= 0, z'' = 1 + (1 + y'), y' >= 0 minus(z', z'') -{ 2 }-> p(0) :|: y >= 0, z'' = 1 + y, z' = 0 minus(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0 p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 ---------------------------------------- (19) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + x2, plus(0, times(x2, 0))) :|: z'' = 1 + x2, x2 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + x1, plus(0, times(x1, 0))) :|: x1 >= 0, z'' = 1 + x1, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + x'', 0)), 0, 0) :|: z'' = 0, z' = 1 + x'', x'' >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + x'', 0)), 1 + x3, plus(1 + x'', times(x3, 1 + x''))) :|: z' = 1 + x'', z'' = 1 + x3, x'' >= 0, x3 >= 0 factorial(z') -{ 1 }-> facIter(x, 1 + 0) :|: z' = x, x >= 0 if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(x, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0 minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 minus(z', z'') -{ 3 }-> x' :|: z' = x, x >= 0, z'' = 1 + 0, x = 1 + x', x' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(x, y'))) :|: z' = x, x >= 0, z'' = 1 + (1 + y'), y' >= 0 minus(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = 1 + 0, x = 0 minus(z', z'') -{ 3 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0, 0 = 0 p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { isZero } { plus } { p } { times } { minus } { if, facIter } { factorial } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {isZero}, {plus}, {p}, {times}, {minus}, {if,facIter}, {factorial} ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {isZero}, {plus}, {p}, {times}, {minus}, {if,facIter}, {factorial} ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {isZero}, {plus}, {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: ?, size: O(1) [1] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) 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'' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (35) 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: 1 + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (39) 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' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {p}, {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z'*z'' + z'' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {times}, {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: ?, size: O(n^2) [2*z'*z'' + z''] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 4*z' + 2*z'*z'' + z'' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 1 + (z'' - 1), plus(0, times(z'' - 1, 0))) :|: z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), plus(1 + (z' - 1), times(z'' - 1, 1 + (z' - 1)))) :|: z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), s6) :|: s5 >= 0, s5 <= 0 + 2 * (0 * (z'' - 1)), s6 >= 0, s6 <= 0 + s5, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, 0, 1 + (z'' - 1), s4) :|: s3 >= 0, s3 <= 0 + 2 * (0 * (z'' - 1)), s4 >= 0, s4 <= 0 + s3, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 5 + 2*z'*z'' + 4*z'' }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), s8) :|: s7 >= 0, s7 <= 1 + (z' - 1) + 2 * ((1 + (z' - 1)) * (z'' - 1)), s8 >= 0, s8 <= 1 + (z' - 1) + s7, z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s2 :|: s'' >= 0, s'' <= z'' + 2 * (z'' * (z' - 2)), s1 >= 0, s1 <= z'' + s'', s2 >= 0, s2 <= z'' + s1, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), s6) :|: s5 >= 0, s5 <= 0 + 2 * (0 * (z'' - 1)), s6 >= 0, s6 <= 0 + s5, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, 0, 1 + (z'' - 1), s4) :|: s3 >= 0, s3 <= 0 + 2 * (0 * (z'' - 1)), s4 >= 0, s4 <= 0 + s3, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 5 + 2*z'*z'' + 4*z'' }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), s8) :|: s7 >= 0, s7 <= 1 + (z' - 1) + 2 * ((1 + (z' - 1)) * (z'' - 1)), s8 >= 0, s8 <= 1 + (z' - 1) + s7, z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s2 :|: s'' >= 0, s'' <= z'' + 2 * (z'' * (z' - 2)), s1 >= 0, s1 <= z'' + s'', s2 >= 0, s2 <= z'' + s1, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {minus}, {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] minus: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 7 + 4*z'' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 4 }-> if(1, p(minus(0, 0)), 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, p(minus(0, 0)), 1 + (z'' - 1), s6) :|: s5 >= 0, s5 <= 0 + 2 * (0 * (z'' - 1)), s6 >= 0, s6 <= 0 + s5, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, 0, 1 + (z'' - 1), s4) :|: s3 >= 0, s3 <= 0 + 2 * (0 * (z'' - 1)), s4 >= 0, s4 <= 0 + s3, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(0, p(minus(1 + (z' - 1), 0)), 0, 0) :|: z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 5 + 2*z'*z'' + 4*z'' }-> if(0, p(minus(1 + (z' - 1), 0)), 1 + (z'' - 1), s8) :|: s7 >= 0, s7 <= 1 + (z' - 1) + 2 * ((1 + (z' - 1)) * (z'' - 1)), s8 >= 0, s8 <= 1 + (z' - 1) + s7, z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 2 }-> p(p(minus(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s2 :|: s'' >= 0, s'' <= z'' + 2 * (z'' * (z' - 2)), s1 >= 0, s1 <= z'' + s'', s2 >= 0, s2 <= z'' + s1, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] minus: runtime: O(n^1) [7 + 4*z''], size: O(n^1) [z'] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 12 }-> if(1, s13, 0, 0) :|: s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12, z'' = 0, z' = 0 facIter(z', z'') -{ 13 + 4*z'' }-> if(1, s15, 1 + (z'' - 1), s6) :|: s14 >= 0, s14 <= 0, s15 >= 0, s15 <= s14, s5 >= 0, s5 <= 0 + 2 * (0 * (z'' - 1)), s6 >= 0, s6 <= 0 + s5, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, 0, 1 + (z'' - 1), s4) :|: s3 >= 0, s3 <= 0 + 2 * (0 * (z'' - 1)), s4 >= 0, s4 <= 0 + s3, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 12 }-> if(0, s17, 0, 0) :|: s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= s16, z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 13 + 2*z'*z'' + 4*z'' }-> if(0, s19, 1 + (z'' - 1), s8) :|: s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s18, s7 >= 0, s7 <= 1 + (z' - 1) + 2 * ((1 + (z' - 1)) * (z'' - 1)), s8 >= 0, s8 <= 1 + (z' - 1) + s7, z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 3 + 4*z'' }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s2 :|: s'' >= 0, s'' <= z'' + 2 * (z'' * (z' - 2)), s1 >= 0, s1 <= z'' + s'', s2 >= 0, s2 <= z'' + s1, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] minus: runtime: O(n^1) [7 + 4*z''], size: O(n^1) [z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) 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: facIter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 12 }-> if(1, s13, 0, 0) :|: s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12, z'' = 0, z' = 0 facIter(z', z'') -{ 13 + 4*z'' }-> if(1, s15, 1 + (z'' - 1), s6) :|: s14 >= 0, s14 <= 0, s15 >= 0, s15 <= s14, s5 >= 0, s5 <= 0 + 2 * (0 * (z'' - 1)), s6 >= 0, s6 <= 0 + s5, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, 0, 1 + (z'' - 1), s4) :|: s3 >= 0, s3 <= 0 + 2 * (0 * (z'' - 1)), s4 >= 0, s4 <= 0 + s3, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 12 }-> if(0, s17, 0, 0) :|: s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= s16, z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 13 + 2*z'*z'' + 4*z'' }-> if(0, s19, 1 + (z'' - 1), s8) :|: s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s18, s7 >= 0, s7 <= 1 + (z' - 1) + 2 * ((1 + (z' - 1)) * (z'' - 1)), s8 >= 0, s8 <= 1 + (z' - 1) + s7, z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 3 + 4*z'' }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s2 :|: s'' >= 0, s'' <= z'' + 2 * (z'' * (z' - 2)), s1 >= 0, s1 <= z'' + s'', s2 >= 0, s2 <= z'' + s1, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] minus: runtime: O(n^1) [7 + 4*z''], size: O(n^1) [z'] if: runtime: ?, size: INF facIter: runtime: ?, size: INF ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: facIter(z', z'') -{ 12 }-> if(1, s13, 0, 0) :|: s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12, z'' = 0, z' = 0 facIter(z', z'') -{ 13 + 4*z'' }-> if(1, s15, 1 + (z'' - 1), s6) :|: s14 >= 0, s14 <= 0, s15 >= 0, s15 <= s14, s5 >= 0, s5 <= 0 + 2 * (0 * (z'' - 1)), s6 >= 0, s6 <= 0 + s5, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 4 }-> if(1, 0, 0, 0) :|: z'' = 0, z' = 0 facIter(z', z'') -{ 5 + 4*z'' }-> if(1, 0, 1 + (z'' - 1), s4) :|: s3 >= 0, s3 <= 0 + 2 * (0 * (z'' - 1)), s4 >= 0, s4 <= 0 + s3, z'' - 1 >= 0, z' = 0 facIter(z', z'') -{ 12 }-> if(0, s17, 0, 0) :|: s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= s16, z'' = 0, z' - 1 >= 0 facIter(z', z'') -{ 13 + 2*z'*z'' + 4*z'' }-> if(0, s19, 1 + (z'' - 1), s8) :|: s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s18, s7 >= 0, s7 <= 1 + (z' - 1) + 2 * ((1 + (z' - 1)) * (z'' - 1)), s8 >= 0, s8 <= 1 + (z' - 1) + s7, z' - 1 >= 0, z'' - 1 >= 0 factorial(z') -{ 1 }-> facIter(z', 1 + 0) :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> facIter(z'', z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 isZero(z') -{ 1 }-> 1 :|: z' = 0 isZero(z') -{ 1 }-> 0 :|: z' - 1 >= 0 minus(z', z'') -{ 3 + 4*z'' }-> s11 :|: s9 >= 0, s9 <= z', s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z' >= 0, z'' - 2 >= 0 minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0 minus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1 + 0, z' = 0 minus(z', z'') -{ 3 }-> 0 :|: z'' - 1 >= 0, z' = 0, 0 = 0 minus(z', z'') -{ 3 }-> z' - 1 :|: z' >= 0, z'' = 1 + 0, z' - 1 >= 0 p(z') -{ 1 }-> 0 :|: z' = 0 p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s' :|: s' >= 0, s' <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s2 :|: s'' >= 0, s'' <= z'' + 2 * (z'' * (z' - 2)), s1 >= 0, s1 <= z'' + s'', s2 >= 0, s2 <= z'' + s1, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 Function symbols to be analyzed: {if,facIter}, {factorial} Previous analysis results are: isZero: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z'] times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z''] minus: runtime: O(n^1) [7 + 4*z''], size: O(n^1) [z'] if: runtime: INF, size: INF facIter: runtime: ?, size: INF ---------------------------------------- (61) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) p(s(z0)) -> z0 p(0) -> 0 minus(z0, 0) -> z0 minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> facIter(z0, z2) factorial(z0) -> facIter(z0, s(0)) Tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(0, z0) -> c2 TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) P(s(z0)) -> c4 P(0) -> c5 MINUS(z0, 0) -> c6 MINUS(0, z0) -> c7 MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1)) ISZERO(0) -> c9 ISZERO(s(z0)) -> c10 FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), ISZERO(z0)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(true, z0, z1, z2) -> c14 IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACTORIAL(z0) -> c16(FACITER(z0, s(0))) S tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(0, z0) -> c2 TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) P(s(z0)) -> c4 P(0) -> c5 MINUS(z0, 0) -> c6 MINUS(0, z0) -> c7 MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1)) ISZERO(0) -> c9 ISZERO(s(z0)) -> c10 FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), ISZERO(z0)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(true, z0, z1, z2) -> c14 IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACTORIAL(z0) -> c16(FACITER(z0, s(0))) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, minus_2, isZero_1, facIter_2, if_4, factorial_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, MINUS_2, ISZERO_1, FACITER_2, IF_4, FACTORIAL_1 Compound Symbols: c, c1_1, c2, c3_2, c4, c5, c6, c7, c8_2, c9, c10, c11_2, c12_2, c13_2, c14, c15_1, c16_1 ---------------------------------------- (63) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FACTORIAL(z0) -> c16(FACITER(z0, s(0))) Removed 9 trailing nodes: ISZERO(s(z0)) -> c10 TIMES(0, z0) -> c2 MINUS(0, z0) -> c7 MINUS(z0, 0) -> c6 P(s(z0)) -> c4 ISZERO(0) -> c9 P(0) -> c5 PLUS(0, z0) -> c IF(true, z0, z1, z2) -> c14 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) p(s(z0)) -> z0 p(0) -> 0 minus(z0, 0) -> z0 minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> facIter(z0, z2) factorial(z0) -> facIter(z0, s(0)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), ISZERO(z0)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), ISZERO(z0)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, minus_2, isZero_1, facIter_2, if_4, factorial_1 Defined Pair Symbols: PLUS_2, TIMES_2, MINUS_2, FACITER_2, IF_4 Compound Symbols: c1_1, c3_2, c8_2, c11_2, c12_2, c13_2, c15_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) p(s(z0)) -> z0 p(0) -> 0 minus(z0, 0) -> z0 minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> facIter(z0, z2) factorial(z0) -> facIter(z0, s(0)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, minus_2, isZero_1, facIter_2, if_4, factorial_1 Defined Pair Symbols: PLUS_2, TIMES_2, FACITER_2, IF_4, MINUS_2 Compound Symbols: c1_1, c3_2, c12_2, c13_2, c15_1, c8_1, c11_1 ---------------------------------------- (67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> facIter(z0, z2) factorial(z0) -> facIter(z0, s(0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) K tuples:none Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, FACITER_2, IF_4, MINUS_2 Compound Symbols: c1_1, c3_2, c12_2, c13_2, c15_1, c8_1, c11_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0))) by FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(true, minus(0, s(0)), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(true, minus(0, s(0)), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(true, minus(0, s(0)), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) K tuples:none Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, FACITER_2, IF_4, MINUS_2 Compound Symbols: c1_1, c3_2, c13_2, c15_1, c8_1, c11_1, c12_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) K tuples:none Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, FACITER_2, IF_4, MINUS_2 Compound Symbols: c1_1, c3_2, c13_2, c15_1, c8_1, c11_1, c12_2, c12_1 ---------------------------------------- (73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACITER(0, x1) -> c12(MINUS(0, s(0))) We considered the (Usable) Rules: isZero(0) -> true isZero(s(z0)) -> false And the Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FACITER(x_1, x_2)) = [1] POL(IF(x_1, x_2, x_3, x_4)) = x_1 POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(false) = [1] POL(isZero(x_1)) = [1] POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(p(x_1)) = [1] + x_1 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, FACITER_2, IF_4, MINUS_2 Compound Symbols: c1_1, c3_2, c13_2, c15_1, c8_1, c11_1, c12_2, c12_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0)) by FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0), TIMES(0, z0)) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(0, x1) -> c13(IF(true, minus(0, s(0)), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0), TIMES(0, z0)) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(0, x1) -> c13(IF(true, minus(0, s(0)), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0), TIMES(0, z0)) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(0, x1) -> c13(IF(true, minus(0, s(0)), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c11_1, c12_2, c12_1, c13_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c11_1, c12_2, c12_1, c13_2, c13_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACITER(0, x1) -> c13(TIMES(x1, 0)) We considered the (Usable) Rules: isZero(0) -> true isZero(s(z0)) -> false And the Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FACITER(x_1, x_2)) = [1] POL(IF(x_1, x_2, x_3, x_4)) = x_1 POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(false) = [1] POL(isZero(x_1)) = [1] POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(p(x_1)) = [1] + x_1 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c11_1, c12_2, c12_1, c13_2, c13_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))) by FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(0, x1) -> c11(IF(true, minus(0, s(0)), x1, times(x1, 0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(0, x1) -> c11(IF(true, minus(0, s(0)), x1, times(x1, 0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(0, x1) -> c11(IF(true, minus(0, s(0)), x1, times(x1, 0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, x1) -> c11(IF(true, minus(0, s(0)), x1, times(x1, 0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, 0) -> c12(IF(isZero(z0), minus(z0, s(0)), 0, 0), MINUS(z0, s(0))) by FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(0, 0) -> c12(IF(true, minus(0, s(0)), 0, 0), MINUS(0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(0, 0) -> c12(IF(true, minus(0, s(0)), 0, 0), MINUS(0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(0, 0) -> c12(IF(true, minus(0, s(0)), 0, 0), MINUS(0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACITER(0, 0) -> c12(MINUS(0, s(0))) We considered the (Usable) Rules:none And the Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FACITER(x_1, x_2)) = [1] POL(IF(x_1, x_2, x_3, x_4)) = [1] POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(false) = [3] POL(isZero(x_1)) = [2] POL(minus(x_1, x_2)) = 0 POL(p(x_1)) = [2] POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = 0 POL(true) = [3] ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z1, s(z0)) -> c12(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) by FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c12(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c12(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c12(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) We considered the (Usable) Rules:none And the Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FACITER(x_1, x_2)) = [1] POL(IF(x_1, x_2, x_3, x_4)) = [1] POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(false) = [1] POL(isZero(x_1)) = [1] POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(p(x_1)) = [1] + x_1 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, x1) -> c12(IF(isZero(0), 0, x1, times(x1, 0)), MINUS(0, s(0))) by FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), plus(0, times(z0, 0))), MINUS(0, s(0))) FACITER(0, x0) -> c12(IF(true, 0, x0, times(x0, 0)), MINUS(0, s(0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x0) -> c12(IF(true, 0, x0, times(x0, 0)), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x0) -> c12(IF(true, 0, x0, times(x0, 0)), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x0) -> c12(MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (101) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x0) -> c12(MINUS(0, s(0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, x1) -> c12(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), MINUS(z0, s(0))) by FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(z1, s(z0)) -> c12(IF(isZero(z1), p(minus(z1, 0)), s(z0), plus(z1, times(z0, z1))), MINUS(z1, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(true, p(minus(0, 0)), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(true, p(minus(0, 0)), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(true, p(minus(0, 0)), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (107) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x0) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_2, c12_1, c13_2, c13_1, c11_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(s(z0), x1) -> c12(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) by FACITER(s(x0), 0) -> c12(IF(false, minus(s(x0), s(0)), 0, 0), MINUS(s(x0), s(0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0)))), MINUS(s(x0), s(0))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0))), MINUS(s(x0), s(0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z1, s(z0)) -> c13(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) by FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(0, s(x1)) -> c13(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(0, s(x1)) -> c13(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(0, s(x1)) -> c13(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) We considered the (Usable) Rules: isZero(0) -> true isZero(s(z0)) -> false And the Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FACITER(x_1, x_2)) = [1] POL(IF(x_1, x_2, x_3, x_4)) = x_1 POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(false) = [1] POL(isZero(x_1)) = [1] POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(p(x_1)) = [1] + x_1 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, x1) -> c13(IF(isZero(0), 0, x1, times(x1, 0)), TIMES(x1, 0)) by FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0), TIMES(0, 0)) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), plus(0, times(z0, 0))), TIMES(s(z0), 0)) FACITER(0, x0) -> c13(IF(true, 0, x0, times(x0, 0)), TIMES(x0, 0)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0), TIMES(0, 0)) FACITER(0, x0) -> c13(IF(true, 0, x0, times(x0, 0)), TIMES(x0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0), TIMES(0, 0)) FACITER(0, x0) -> c13(IF(true, 0, x0, times(x0, 0)), TIMES(x0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x0) -> c13(TIMES(x0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (121) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, x0) -> c13(TIMES(x0, 0)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, x1) -> c13(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0)), TIMES(x1, z0)) by FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0), TIMES(0, z0)) FACITER(z1, s(z0)) -> c13(IF(isZero(z1), p(minus(z1, 0)), s(z0), plus(z1, times(z0, z1))), TIMES(s(z0), z1)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(0, x1) -> c13(IF(true, p(minus(0, 0)), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0), TIMES(0, z0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(0, x1) -> c13(IF(true, p(minus(0, 0)), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0), TIMES(0, z0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(0, x1) -> c13(IF(true, p(minus(0, 0)), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (127) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, x0) -> c13(TIMES(x0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_2, c13_1, c11_1, c12_2 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(s(z0), x1) -> c13(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) by FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0), TIMES(0, s(x0))) FACITER(s(x0), s(z0)) -> c13(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0)))), TIMES(s(z0), s(x0))) FACITER(s(x0), x1) -> c13(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0))), TIMES(x1, s(x0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0), TIMES(0, s(x0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0), TIMES(0, s(x0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, 0) -> c13(IF(isZero(z0), minus(z0, s(0)), 0, 0)) by FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, 0) -> c13(IF(true, minus(0, s(0)), 0, 0)) FACITER(s(z0), 0) -> c13(IF(false, minus(s(z0), s(0)), 0, 0)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c13(IF(true, minus(0, s(0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c13(IF(true, minus(0, s(0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, 0) -> c13(IF(true, minus(0, s(0)), 0, 0)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, 0) -> c11(IF(isZero(z0), minus(z0, s(0)), 0, 0)) by FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, 0) -> c11(IF(true, minus(0, s(0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, 0) -> c11(IF(true, minus(0, s(0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, 0) -> c11(IF(true, minus(0, s(0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, 0) -> c11(IF(true, minus(0, s(0)), 0, 0)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z1, s(z0)) -> c11(IF(isZero(z1), minus(z1, s(0)), s(z0), plus(z1, times(z0, z1)))) by FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(0, s(x1)) -> c11(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(0, s(x1)) -> c11(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(0, s(x1)) -> c11(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, s(x1)) -> c11(IF(true, minus(0, s(0)), s(x1), plus(0, times(x1, 0)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, x1) -> c11(IF(isZero(0), 0, x1, times(x1, 0))) by FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(0, s(z0)) -> c11(IF(isZero(0), 0, s(z0), plus(0, times(z0, 0)))) FACITER(0, x0) -> c11(IF(true, 0, x0, times(x0, 0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x0) -> c11(IF(true, 0, x0, times(x0, 0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x0) -> c11(IF(true, 0, x0, times(x0, 0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, x0) -> c11(IF(true, 0, x0, times(x0, 0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(z0, x1) -> c11(IF(isZero(z0), p(minus(z0, 0)), x1, times(x1, z0))) by FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(z1, s(z0)) -> c11(IF(isZero(z1), p(minus(z1, 0)), s(z0), plus(z1, times(z0, z1)))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(0, x1) -> c11(IF(true, p(minus(0, 0)), x1, times(x1, 0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(0, x1) -> c11(IF(true, p(minus(0, 0)), x1, times(x1, 0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(0, x1) -> c11(IF(true, p(minus(0, 0)), x1, times(x1, 0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, x1) -> c11(IF(true, p(minus(0, 0)), x1, times(x1, 0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c11_1, c12_2, c13_2 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(s(z0), x1) -> c11(IF(false, minus(s(z0), s(0)), x1, times(x1, s(z0)))) by FACITER(s(x0), 0) -> c11(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(s(x0), s(z0)) -> c11(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c11(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) by FACITER(0, 0) -> c12(IF(true, 0, 0, 0), MINUS(0, s(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(true, 0, 0, 0), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(true, 0, 0, 0), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (159) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, 0) -> c12(IF(isZero(0), 0, 0, 0), MINUS(0, s(0))) by FACITER(0, 0) -> c12(IF(true, 0, 0, 0), MINUS(0, s(0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c12(IF(true, 0, 0, 0), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) by FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, 0) -> c13(IF(isZero(0), 0, 0, 0)) by FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, 0) -> c13(IF(true, 0, 0, 0)) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) by FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACITER(0, 0) -> c11(IF(isZero(0), 0, 0, 0)) by FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (179) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, 0) -> c11(IF(true, 0, 0, 0)) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(x1)) -> c12(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), MINUS(0, s(0))) by FACITER(0, s(z0)) -> c12(IF(true, minus(0, s(0)), s(z0), times(z0, 0)), MINUS(0, s(0))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, s(z0)) -> c12(IF(true, minus(0, s(0)), s(z0), times(z0, 0)), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, s(z0)) -> c12(IF(true, minus(0, s(0)), s(z0), times(z0, 0)), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (183) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(0, s(z0)) -> c12(MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (185) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, s(z0)) -> c12(MINUS(0, s(0))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), s(x1)) -> c12(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), MINUS(s(z0), s(0))) by FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) by FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (191) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) by FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(0, s(x1)) -> c12(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), MINUS(0, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), plus(0, times(z0, 0))), MINUS(0, s(0))) by FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, x1) -> c12(IF(isZero(0), p(0), x1, times(x1, 0)), MINUS(0, s(0))) by FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (197) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) by FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: PLUS_2, TIMES_2, IF_4, MINUS_2, FACITER_2 Compound Symbols: c1_1, c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) by PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, MINUS_2, FACITER_2, PLUS_2 Compound Symbols: c3_2, c15_1, c8_1, c12_1, c13_1, c12_2, c13_2, c11_1, c1_1 ---------------------------------------- (201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(z0, s(z1)) -> c8(MINUS(z0, z1)) by MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(s(z0), s(x1)) -> c12(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(z0), x1) -> c12(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), MINUS(s(z0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) K tuples: FACITER(0, x1) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(x1)) -> c12(MINUS(0, s(0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c12_1, c13_1, c12_2, c13_2, c11_1, c1_1, c8_1 ---------------------------------------- (203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: FACITER(0, 0) -> c12(MINUS(0, s(0))) FACITER(0, s(z0)) -> c12(MINUS(0, s(0))) FACITER(0, x1) -> c12(MINUS(0, s(0))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0)))), MINUS(s(x0), s(0))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0))), MINUS(s(x0), s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0), MINUS(z0, s(0))) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0), MINUS(s(z0), s(0))) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), MINUS(z0, s(0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), MINUS(z1, s(0))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), MINUS(z0, s(0))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0)), MINUS(z0, s(0))) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), MINUS(s(z0), s(0))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0)), MINUS(0, s(0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0)), MINUS(0, s(0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0))), MINUS(s(z0), s(0))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c12_2, c13_2, c11_1, c1_1, c8_1 ---------------------------------------- (205) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 17 trailing tuple parts ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (207) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) We considered the (Usable) Rules:none And the Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FACITER(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = 0 POL(MINUS(x_1, x_2)) = [3]x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(false) = [3] POL(isZero(x_1)) = [2] POL(minus(x_1, x_2)) = 0 POL(p(x_1)) = [2] POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = [2] + x_1 POL(times(x_1, x_2)) = 0 POL(true) = [3] ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (209) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(x1)) -> c13(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0)), TIMES(s(x1), 0)) by FACITER(0, s(z0)) -> c13(IF(true, minus(0, s(0)), s(z0), times(z0, 0)), TIMES(s(z0), 0)) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(0, s(z0)) -> c13(IF(true, minus(0, s(0)), s(z0), times(z0, 0)), TIMES(s(z0), 0)) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(0, s(z0)) -> c13(IF(true, minus(0, s(0)), s(z0), times(z0, 0)), TIMES(s(z0), 0)) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (211) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(0, s(z0)) -> c13(TIMES(s(z0), 0)) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (213) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, s(z0)) -> c13(TIMES(s(z0), 0)) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (215) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), s(x1)) -> c13(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0))))), TIMES(s(x1), s(z0))) by FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (217) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) by FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (219) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) by FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(0, s(x1)) -> c13(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0))), TIMES(s(x1), 0)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (221) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), plus(0, times(z0, 0))), TIMES(s(z0), 0)) by FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (223) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, x1) -> c13(IF(isZero(0), p(0), x1, times(x1, 0)), TIMES(x1, 0)) by FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (225) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) by FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) K tuples: FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (227) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FACITER(0, x1) -> c13(TIMES(x1, 0)) by FACITER(0, s(y0)) -> c13(TIMES(s(y0), 0)) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(s(z0), s(x1)) -> c13(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0)))), TIMES(s(x1), s(z0))) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (229) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(x0), s(z0)) -> c13(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0)))), TIMES(s(z0), s(x0))) by FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(s(z0), x1) -> c13(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0))), TIMES(x1, s(z0))) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (231) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(x0), x1) -> c13(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0))), TIMES(x1, s(x0))) by FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (233) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FACITER(0, x0) -> c13(TIMES(x0, 0)) by FACITER(0, s(y0)) -> c13(TIMES(s(y0), 0)) ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (235) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(0, s(x1)) -> c11(IF(isZero(0), minus(0, s(0)), s(x1), times(x1, 0))) by FACITER(0, s(z0)) -> c11(IF(true, minus(0, s(0)), s(z0), times(z0, 0))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) FACITER(0, s(z0)) -> c11(IF(true, minus(0, s(0)), s(z0), times(z0, 0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) FACITER(0, s(z0)) -> c11(IF(true, minus(0, s(0)), s(z0), times(z0, 0))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_1, c13_2, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (237) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACITER(0, s(z0)) -> c11(IF(true, minus(0, s(0)), s(z0), times(z0, 0))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_2, c13_1, c11_1, c1_1, c8_1, c12_1 ---------------------------------------- (239) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACITER(s(z0), s(x1)) -> c11(IF(isZero(s(z0)), minus(s(z0), s(0)), s(x1), s(plus(z0, times(x1, s(z0)))))) by FACITER(s(z0), s(z1)) -> c11(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) isZero(0) -> true isZero(s(z0)) -> false minus(0, z0) -> 0 minus(z0, s(z1)) -> p(minus(z0, z1)) minus(z0, 0) -> z0 p(s(z0)) -> z0 p(0) -> 0 Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(0, x1) -> c13(TIMES(x1, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(x0), s(z0)) -> c12(IF(false, minus(s(x0), s(0)), s(z0), plus(s(x0), times(z0, s(x0))))) FACITER(s(x0), x1) -> c12(IF(false, p(minus(s(x0), 0)), x1, times(x1, s(x0)))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) FACITER(s(z0), s(z1)) -> c11(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2) -> c15(FACITER(z0, z2)) FACITER(z0, s(0)) -> c13(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0)), TIMES(s(0), z0)) FACITER(z1, s(s(z0))) -> c13(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1)))), TIMES(s(s(z0)), z1)) FACITER(z0, s(x1)) -> c13(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0))), TIMES(s(x1), z0)) FACITER(z0, x1) -> c13(IF(isZero(z0), p(z0), x1, times(x1, z0)), TIMES(x1, z0)) FACITER(z0, 0) -> c13(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(x0), 0) -> c13(IF(false, minus(s(x0), s(0)), 0, 0)) FACITER(z0, 0) -> c11(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c11(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c11(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c11(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(0, s(x1)) -> c11(IF(isZero(0), 0, s(x1), plus(0, times(x1, 0)))) FACITER(z0, s(x1)) -> c11(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(s(z0), s(x1)) -> c11(IF(false, minus(s(z0), s(0)), s(x1), plus(s(z0), times(x1, s(z0))))) FACITER(0, x1) -> c11(IF(isZero(0), p(0), x1, times(x1, 0))) FACITER(z0, x1) -> c11(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), x1) -> c11(IF(false, p(minus(s(z0), 0)), x1, times(x1, s(z0)))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FACITER(z0, 0) -> c12(IF(isZero(z0), p(minus(z0, 0)), 0, 0)) FACITER(s(z0), 0) -> c12(IF(false, minus(s(z0), s(0)), 0, 0)) FACITER(z0, s(0)) -> c12(IF(isZero(z0), minus(z0, s(0)), s(0), plus(z0, 0))) FACITER(z1, s(s(z0))) -> c12(IF(isZero(z1), minus(z1, s(0)), s(s(z0)), plus(z1, plus(z1, times(z0, z1))))) FACITER(z0, s(x1)) -> c12(IF(isZero(z0), p(minus(z0, 0)), s(x1), plus(z0, times(x1, z0)))) FACITER(z0, x1) -> c12(IF(isZero(z0), p(z0), x1, times(x1, z0))) FACITER(s(z0), s(z1)) -> c12(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) FACITER(0, s(z0)) -> c12(IF(isZero(0), 0, s(z0), times(z0, 0))) FACITER(0, z0) -> c12(IF(isZero(0), 0, z0, times(z0, 0))) FACITER(s(z0), z1) -> c12(IF(false, p(s(z0)), z1, times(z1, s(z0)))) FACITER(s(z0), s(z1)) -> c13(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0))))), TIMES(s(z1), s(z0))) FACITER(0, s(z0)) -> c13(IF(isZero(0), 0, s(z0), times(z0, 0)), TIMES(s(z0), 0)) FACITER(0, z0) -> c13(IF(isZero(0), 0, z0, times(z0, 0)), TIMES(z0, 0)) FACITER(s(z0), z1) -> c13(IF(false, p(s(z0)), z1, times(z1, s(z0))), TIMES(z1, s(z0))) FACITER(s(z0), s(z1)) -> c11(IF(false, minus(s(z0), s(0)), s(z1), s(plus(z0, times(z1, s(z0)))))) K tuples: FACITER(x0, s(x1)) -> c13(TIMES(s(x1), x0)) FACITER(0, s(x1)) -> c13(TIMES(s(x1), 0)) MINUS(z0, s(s(y1))) -> c8(MINUS(z0, s(y1))) Defined Rule Symbols: times_2, plus_2, isZero_1, minus_2, p_1 Defined Pair Symbols: TIMES_2, IF_4, FACITER_2, PLUS_2, MINUS_2 Compound Symbols: c3_2, c15_1, c13_2, c13_1, c11_1, c1_1, c8_1, c12_1