KILLED proof of input_uhRUQBEaan.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 578 ms] (2) CpxRelTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 19 ms] (16) CpxRNTS (17) InliningProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 3 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 3 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 193 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 408 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 224 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 5 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 161 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 2834 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 403 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 628 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 173 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 567 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (70) CpxRNTS (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (76) CpxRNTS (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (82) CpxRNTS (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (84) CpxRNTS (85) IntTrsBoundProof [UPPER BOUND(ID), 6782 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 289 ms] (88) CpxRNTS (89) CompletionProof [UPPER BOUND(ID), 0 ms] (90) CpxTypedWeightedCompleteTrs (91) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 14 ms] (92) CpxRNTS (93) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 13 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 116 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 170 ms] (108) CdtProblem (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 4070 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 110 ms] (118) CdtProblem (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 159 ms] (120) CdtProblem (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 156 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 149 ms] (124) CdtProblem (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 157 ms] (126) CdtProblem (127) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 1359 ms] (128) CdtProblem (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 393 ms] (130) CdtProblem (131) CdtKnowledgeProof [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), 1 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (148) CdtProblem (149) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 18 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 17 ms] (160) CdtProblem (161) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 9 ms] (162) CdtProblem (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (180) CdtProblem (181) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (200) CdtProblem (201) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (204) CdtProblem (205) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (208) CdtProblem (209) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 7 ms] (214) CdtProblem (215) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 27 ms] (218) CdtProblem (219) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (242) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: head(Cons(x, xs)) -> x factor(Cons(RPar, xs)) -> xs factor(Cons(Div, xs)) -> xs factor(Cons(Mul, xs)) -> xs factor(Cons(Plus, xs)) -> xs factor(Cons(Minus, xs)) -> xs factor(Cons(Val(int), xs)) -> xs factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) member(x, Nil) -> False atom(Cons(x, xs)) -> xs atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(int2)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(int2)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(int2)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(int2)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(int2)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(int2)) -> False eqAlph(Val(int), RPar) -> False eqAlph(Val(int), LPar) -> False eqAlph(Val(int), Div) -> False eqAlph(Val(int), Mul) -> False eqAlph(Val(int), Plus) -> False eqAlph(Val(int), Minus) -> False eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False term(xs) -> term[Let](xs, factor(xs)) parsexp(xs) -> expr(xs) expr(xs) -> expr[Let](xs, term(xs)) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(x), S(y)) -> !EQ(x, y) !EQ(0, S(y)) -> False !EQ(S(x), 0) -> False !EQ(0, 0) -> True factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) term[Let](xs, Nil) -> Nil member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) expr[Let](xs, Nil) -> Nil member[Ite][True][Ite](True, x, xs) -> True Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: head(Cons(x, xs)) -> x factor(Cons(RPar, xs)) -> xs factor(Cons(Div, xs)) -> xs factor(Cons(Mul, xs)) -> xs factor(Cons(Plus, xs)) -> xs factor(Cons(Minus, xs)) -> xs factor(Cons(Val(int), xs)) -> xs factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) member(x, Nil) -> False atom(Cons(x, xs)) -> xs atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(int2)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(int2)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(int2)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(int2)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(int2)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(int2)) -> False eqAlph(Val(int), RPar) -> False eqAlph(Val(int), LPar) -> False eqAlph(Val(int), Div) -> False eqAlph(Val(int), Mul) -> False eqAlph(Val(int), Plus) -> False eqAlph(Val(int), Minus) -> False eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False term(xs) -> term[Let](xs, factor(xs)) parsexp(xs) -> expr(xs) expr(xs) -> expr[Let](xs, term(xs)) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(x), S(y)) -> !EQ(x, y) !EQ(0, S(y)) -> False !EQ(S(x), 0) -> False !EQ(0, 0) -> True factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) term[Let](xs, Nil) -> Nil member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) expr[Let](xs, Nil) -> Nil member[Ite][True][Ite](True, x, xs) -> True 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: head(Cons(x, xs)) -> x factor(Cons(RPar, xs)) -> xs factor(Cons(Div, xs)) -> xs factor(Cons(Mul, xs)) -> xs factor(Cons(Plus, xs)) -> xs factor(Cons(Minus, xs)) -> xs factor(Cons(Val(int), xs)) -> xs factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) member(x, Nil) -> False atom(Cons(x, xs)) -> xs atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(int2)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(int2)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(int2)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(int2)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(int2)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(int2)) -> False eqAlph(Val(int), RPar) -> False eqAlph(Val(int), LPar) -> False eqAlph(Val(int), Div) -> False eqAlph(Val(int), Mul) -> False eqAlph(Val(int), Plus) -> False eqAlph(Val(int), Minus) -> False eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False term(xs) -> term[Let](xs, factor(xs)) parsexp(xs) -> expr(xs) expr(xs) -> expr[Let](xs, term(xs)) and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(x), S(y)) -> !EQ(x, y) !EQ(0, S(y)) -> False !EQ(S(x), 0) -> False !EQ(0, 0) -> True factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) term[Let](xs, Nil) -> Nil member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) expr[Let](xs, Nil) -> Nil member[Ite][True][Ite](True, x, xs) -> True S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: head(Cons(x, xs)) -> x factor(Cons(RPar, xs)) -> xs factor(Cons(Div, xs)) -> xs factor(Cons(Mul, xs)) -> xs factor(Cons(Plus, xs)) -> xs factor(Cons(Minus, xs)) -> xs factor(Cons(Val(int), xs)) -> xs factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) member(x, Nil) -> False atom(Cons(x, xs)) -> xs atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(int2)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(int2)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(int2)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(int2)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(int2)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(int2)) -> False eqAlph(Val(int), RPar) -> False eqAlph(Val(int), LPar) -> False eqAlph(Val(int), Div) -> False eqAlph(Val(int), Mul) -> False eqAlph(Val(int), Plus) -> False eqAlph(Val(int), Minus) -> False eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False term(xs) -> term[Let](xs, factor(xs)) parsexp(xs) -> expr(xs) expr(xs) -> expr[Let](xs, term(xs)) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(x), S(y)) -> !EQ(x, y) !EQ(0', S(y)) -> False !EQ(S(x), 0') -> False !EQ(0', 0') -> True factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) term[Let](xs, Nil) -> Nil member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) expr[Let](xs, Nil) -> Nil member[Ite][True][Ite](True, x, xs) -> True Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (7) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (8) 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: head(Cons(x, xs)) -> x [1] factor(Cons(RPar, xs)) -> xs [1] factor(Cons(Div, xs)) -> xs [1] factor(Cons(Mul, xs)) -> xs [1] factor(Cons(Plus, xs)) -> xs [1] factor(Cons(Minus, xs)) -> xs [1] factor(Cons(Val(int), xs)) -> xs [1] factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] atom(Cons(x, xs)) -> xs [1] atom(Nil) -> Nil [1] eqAlph(RPar, RPar) -> True [1] eqAlph(RPar, LPar) -> False [1] eqAlph(RPar, Div) -> False [1] eqAlph(RPar, Mul) -> False [1] eqAlph(RPar, Plus) -> False [1] eqAlph(RPar, Minus) -> False [1] eqAlph(RPar, Val(int2)) -> False [1] eqAlph(LPar, RPar) -> False [1] eqAlph(LPar, LPar) -> True [1] eqAlph(LPar, Div) -> False [1] eqAlph(LPar, Mul) -> False [1] eqAlph(LPar, Plus) -> False [1] eqAlph(LPar, Minus) -> False [1] eqAlph(LPar, Val(int2)) -> False [1] eqAlph(Div, RPar) -> False [1] eqAlph(Div, LPar) -> False [1] eqAlph(Div, Div) -> True [1] eqAlph(Div, Mul) -> False [1] eqAlph(Div, Plus) -> False [1] eqAlph(Div, Minus) -> False [1] eqAlph(Div, Val(int2)) -> False [1] eqAlph(Mul, RPar) -> False [1] eqAlph(Mul, LPar) -> False [1] eqAlph(Mul, Div) -> False [1] eqAlph(Mul, Mul) -> True [1] eqAlph(Mul, Plus) -> False [1] eqAlph(Mul, Minus) -> False [1] eqAlph(Mul, Val(int2)) -> False [1] eqAlph(Plus, RPar) -> False [1] eqAlph(Plus, LPar) -> False [1] eqAlph(Plus, Div) -> False [1] eqAlph(Plus, Mul) -> False [1] eqAlph(Plus, Plus) -> True [1] eqAlph(Plus, Minus) -> False [1] eqAlph(Plus, Val(int2)) -> False [1] eqAlph(Minus, RPar) -> False [1] eqAlph(Minus, LPar) -> False [1] eqAlph(Minus, Div) -> False [1] eqAlph(Minus, Mul) -> False [1] eqAlph(Minus, Plus) -> False [1] eqAlph(Minus, Minus) -> True [1] eqAlph(Minus, Val(int2)) -> False [1] eqAlph(Val(int), RPar) -> False [1] eqAlph(Val(int), LPar) -> False [1] eqAlph(Val(int), Div) -> False [1] eqAlph(Val(int), Mul) -> False [1] eqAlph(Val(int), Plus) -> False [1] eqAlph(Val(int), Minus) -> False [1] eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] term(xs) -> term[Let](xs, factor(xs)) [1] parsexp(xs) -> expr(xs) [1] expr(xs) -> expr[Let](xs, term(xs)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) [0] factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) [0] factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) [0] factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) [0] factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) [0] factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) [0] factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) [0] term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) [0] expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) [0] term[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) [0] expr[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](True, x, xs) -> True [0] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: head(Cons(x, xs)) -> x [1] factor(Cons(RPar, xs)) -> xs [1] factor(Cons(Div, xs)) -> xs [1] factor(Cons(Mul, xs)) -> xs [1] factor(Cons(Plus, xs)) -> xs [1] factor(Cons(Minus, xs)) -> xs [1] factor(Cons(Val(int), xs)) -> xs [1] factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] atom(Cons(x, xs)) -> xs [1] atom(Nil) -> Nil [1] eqAlph(RPar, RPar) -> True [1] eqAlph(RPar, LPar) -> False [1] eqAlph(RPar, Div) -> False [1] eqAlph(RPar, Mul) -> False [1] eqAlph(RPar, Plus) -> False [1] eqAlph(RPar, Minus) -> False [1] eqAlph(RPar, Val(int2)) -> False [1] eqAlph(LPar, RPar) -> False [1] eqAlph(LPar, LPar) -> True [1] eqAlph(LPar, Div) -> False [1] eqAlph(LPar, Mul) -> False [1] eqAlph(LPar, Plus) -> False [1] eqAlph(LPar, Minus) -> False [1] eqAlph(LPar, Val(int2)) -> False [1] eqAlph(Div, RPar) -> False [1] eqAlph(Div, LPar) -> False [1] eqAlph(Div, Div) -> True [1] eqAlph(Div, Mul) -> False [1] eqAlph(Div, Plus) -> False [1] eqAlph(Div, Minus) -> False [1] eqAlph(Div, Val(int2)) -> False [1] eqAlph(Mul, RPar) -> False [1] eqAlph(Mul, LPar) -> False [1] eqAlph(Mul, Div) -> False [1] eqAlph(Mul, Mul) -> True [1] eqAlph(Mul, Plus) -> False [1] eqAlph(Mul, Minus) -> False [1] eqAlph(Mul, Val(int2)) -> False [1] eqAlph(Plus, RPar) -> False [1] eqAlph(Plus, LPar) -> False [1] eqAlph(Plus, Div) -> False [1] eqAlph(Plus, Mul) -> False [1] eqAlph(Plus, Plus) -> True [1] eqAlph(Plus, Minus) -> False [1] eqAlph(Plus, Val(int2)) -> False [1] eqAlph(Minus, RPar) -> False [1] eqAlph(Minus, LPar) -> False [1] eqAlph(Minus, Div) -> False [1] eqAlph(Minus, Mul) -> False [1] eqAlph(Minus, Plus) -> False [1] eqAlph(Minus, Minus) -> True [1] eqAlph(Minus, Val(int2)) -> False [1] eqAlph(Val(int), RPar) -> False [1] eqAlph(Val(int), LPar) -> False [1] eqAlph(Val(int), Div) -> False [1] eqAlph(Val(int), Mul) -> False [1] eqAlph(Val(int), Plus) -> False [1] eqAlph(Val(int), Minus) -> False [1] eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] term(xs) -> term[Let](xs, factor(xs)) [1] parsexp(xs) -> expr(xs) [1] expr(xs) -> expr[Let](xs, term(xs)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) [0] factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) [0] factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) [0] factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) [0] factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) [0] factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) [0] factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) [0] term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) [0] expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) [0] term[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) [0] expr[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](True, x, xs) -> True [0] The TRS has the following type information: head :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> RPar:Div:Mul:Plus:Minus:Val:LPar Cons :: RPar:Div:Mul:Plus:Minus:Val:LPar -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] factor :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] RPar :: RPar:Div:Mul:Plus:Minus:Val:LPar Div :: RPar:Div:Mul:Plus:Minus:Val:LPar Mul :: RPar:Div:Mul:Plus:Minus:Val:LPar Plus :: RPar:Div:Mul:Plus:Minus:Val:LPar Minus :: RPar:Div:Mul:Plus:Minus:Val:LPar Val :: S:0 -> RPar:Div:Mul:Plus:Minus:Val:LPar LPar :: RPar:Div:Mul:Plus:Minus:Val:LPar factor[Ite][True][Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] member :: RPar:Div:Mul:Plus:Minus:Val:LPar -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True member[Ite][True][Ite] :: False:True -> RPar:Div:Mul:Plus:Minus:Val:LPar -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True eqAlph :: RPar:Div:Mul:Plus:Minus:Val:LPar -> RPar:Div:Mul:Plus:Minus:Val:LPar -> False:True Nil :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] False :: False:True atom :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] True :: False:True !EQ :: S:0 -> S:0 -> False:True notEmpty :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True term :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] term[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] parsexp :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] and :: False:True -> False:True -> False:True S :: S:0 -> S:0 0 :: S:0 factor[Ite][True][Let][Ite] :: False:True -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] term[Let][Ite][False][Ite] :: False:True -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr[Let][Ite][False][Ite] :: False:True -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: atom_1 notEmpty_1 parsexp_1 (c) The following functions are completely defined: eqAlph_2 factor_1 expr_1 term_1 head_1 member_2 and_2 !EQ_2 factor[Ite][True][Let]_2 term[Let]_2 expr[Let]_2 member[Ite][True][Ite]_3 Due to the following rules being added: and(v0, v1) -> null_and [0] !EQ(v0, v1) -> null_!EQ [0] factor[Ite][True][Let](v0, v1) -> Nil [0] term[Let](v0, v1) -> Nil [0] expr[Let](v0, v1) -> Nil [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] factor(v0) -> Nil [0] head(v0) -> null_head [0] member(v0, v1) -> null_member [0] eqAlph(v0, v1) -> null_eqAlph [0] And the following fresh constants: null_and, null_!EQ, null_member[Ite][True][Ite], null_head, null_member, null_eqAlph ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: head(Cons(x, xs)) -> x [1] factor(Cons(RPar, xs)) -> xs [1] factor(Cons(Div, xs)) -> xs [1] factor(Cons(Mul, xs)) -> xs [1] factor(Cons(Plus, xs)) -> xs [1] factor(Cons(Minus, xs)) -> xs [1] factor(Cons(Val(int), xs)) -> xs [1] factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] atom(Cons(x, xs)) -> xs [1] atom(Nil) -> Nil [1] eqAlph(RPar, RPar) -> True [1] eqAlph(RPar, LPar) -> False [1] eqAlph(RPar, Div) -> False [1] eqAlph(RPar, Mul) -> False [1] eqAlph(RPar, Plus) -> False [1] eqAlph(RPar, Minus) -> False [1] eqAlph(RPar, Val(int2)) -> False [1] eqAlph(LPar, RPar) -> False [1] eqAlph(LPar, LPar) -> True [1] eqAlph(LPar, Div) -> False [1] eqAlph(LPar, Mul) -> False [1] eqAlph(LPar, Plus) -> False [1] eqAlph(LPar, Minus) -> False [1] eqAlph(LPar, Val(int2)) -> False [1] eqAlph(Div, RPar) -> False [1] eqAlph(Div, LPar) -> False [1] eqAlph(Div, Div) -> True [1] eqAlph(Div, Mul) -> False [1] eqAlph(Div, Plus) -> False [1] eqAlph(Div, Minus) -> False [1] eqAlph(Div, Val(int2)) -> False [1] eqAlph(Mul, RPar) -> False [1] eqAlph(Mul, LPar) -> False [1] eqAlph(Mul, Div) -> False [1] eqAlph(Mul, Mul) -> True [1] eqAlph(Mul, Plus) -> False [1] eqAlph(Mul, Minus) -> False [1] eqAlph(Mul, Val(int2)) -> False [1] eqAlph(Plus, RPar) -> False [1] eqAlph(Plus, LPar) -> False [1] eqAlph(Plus, Div) -> False [1] eqAlph(Plus, Mul) -> False [1] eqAlph(Plus, Plus) -> True [1] eqAlph(Plus, Minus) -> False [1] eqAlph(Plus, Val(int2)) -> False [1] eqAlph(Minus, RPar) -> False [1] eqAlph(Minus, LPar) -> False [1] eqAlph(Minus, Div) -> False [1] eqAlph(Minus, Mul) -> False [1] eqAlph(Minus, Plus) -> False [1] eqAlph(Minus, Minus) -> True [1] eqAlph(Minus, Val(int2)) -> False [1] eqAlph(Val(int), RPar) -> False [1] eqAlph(Val(int), LPar) -> False [1] eqAlph(Val(int), Div) -> False [1] eqAlph(Val(int), Mul) -> False [1] eqAlph(Val(int), Plus) -> False [1] eqAlph(Val(int), Minus) -> False [1] eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] term(xs) -> term[Let](xs, factor(xs)) [1] parsexp(xs) -> expr(xs) [1] expr(xs) -> expr[Let](xs, term(xs)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) [0] factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) [0] factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) [0] factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) [0] factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) [0] factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) [0] factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) [0] term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) [0] expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) [0] term[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) [0] expr[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](True, x, xs) -> True [0] and(v0, v1) -> null_and [0] !EQ(v0, v1) -> null_!EQ [0] factor[Ite][True][Let](v0, v1) -> Nil [0] term[Let](v0, v1) -> Nil [0] expr[Let](v0, v1) -> Nil [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] factor(v0) -> Nil [0] head(v0) -> null_head [0] member(v0, v1) -> null_member [0] eqAlph(v0, v1) -> null_eqAlph [0] The TRS has the following type information: head :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Cons :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] factor :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] RPar :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Div :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Mul :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Plus :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Minus :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Val :: S:0 -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head LPar :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head factor[Ite][True][Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] member :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph member[Ite][True][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph eqAlph :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph Nil :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] False :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph atom :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] True :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph !EQ :: S:0 -> S:0 -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph notEmpty :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph term :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] term[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] parsexp :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] and :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph S :: S:0 -> S:0 0 :: S:0 factor[Ite][True][Let][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] term[Let][Ite][False][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr[Let][Ite][False][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] null_and :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_!EQ :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_member[Ite][True][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_head :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head null_member :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_eqAlph :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: head(Cons(x, xs)) -> x [1] factor(Cons(RPar, xs)) -> xs [1] factor(Cons(Div, xs)) -> xs [1] factor(Cons(Mul, xs)) -> xs [1] factor(Cons(Plus, xs)) -> xs [1] factor(Cons(Minus, xs)) -> xs [1] factor(Cons(Val(int), xs)) -> xs [1] factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr[Let](Cons(LPar, xs), term(Cons(LPar, xs)))) [2] member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] atom(Cons(x, xs)) -> xs [1] atom(Nil) -> Nil [1] eqAlph(RPar, RPar) -> True [1] eqAlph(RPar, LPar) -> False [1] eqAlph(RPar, Div) -> False [1] eqAlph(RPar, Mul) -> False [1] eqAlph(RPar, Plus) -> False [1] eqAlph(RPar, Minus) -> False [1] eqAlph(RPar, Val(int2)) -> False [1] eqAlph(LPar, RPar) -> False [1] eqAlph(LPar, LPar) -> True [1] eqAlph(LPar, Div) -> False [1] eqAlph(LPar, Mul) -> False [1] eqAlph(LPar, Plus) -> False [1] eqAlph(LPar, Minus) -> False [1] eqAlph(LPar, Val(int2)) -> False [1] eqAlph(Div, RPar) -> False [1] eqAlph(Div, LPar) -> False [1] eqAlph(Div, Div) -> True [1] eqAlph(Div, Mul) -> False [1] eqAlph(Div, Plus) -> False [1] eqAlph(Div, Minus) -> False [1] eqAlph(Div, Val(int2)) -> False [1] eqAlph(Mul, RPar) -> False [1] eqAlph(Mul, LPar) -> False [1] eqAlph(Mul, Div) -> False [1] eqAlph(Mul, Mul) -> True [1] eqAlph(Mul, Plus) -> False [1] eqAlph(Mul, Minus) -> False [1] eqAlph(Mul, Val(int2)) -> False [1] eqAlph(Plus, RPar) -> False [1] eqAlph(Plus, LPar) -> False [1] eqAlph(Plus, Div) -> False [1] eqAlph(Plus, Mul) -> False [1] eqAlph(Plus, Plus) -> True [1] eqAlph(Plus, Minus) -> False [1] eqAlph(Plus, Val(int2)) -> False [1] eqAlph(Minus, RPar) -> False [1] eqAlph(Minus, LPar) -> False [1] eqAlph(Minus, Div) -> False [1] eqAlph(Minus, Mul) -> False [1] eqAlph(Minus, Plus) -> False [1] eqAlph(Minus, Minus) -> True [1] eqAlph(Minus, Val(int2)) -> False [1] eqAlph(Val(int), RPar) -> False [1] eqAlph(Val(int), LPar) -> False [1] eqAlph(Val(int), Div) -> False [1] eqAlph(Val(int), Mul) -> False [1] eqAlph(Val(int), Plus) -> False [1] eqAlph(Val(int), Minus) -> False [1] eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] term(Cons(RPar, xs'')) -> term[Let](Cons(RPar, xs''), xs'') [2] term(Cons(Div, xs1)) -> term[Let](Cons(Div, xs1), xs1) [2] term(Cons(Mul, xs2)) -> term[Let](Cons(Mul, xs2), xs2) [2] term(Cons(Plus, xs3)) -> term[Let](Cons(Plus, xs3), xs3) [2] term(Cons(Minus, xs4)) -> term[Let](Cons(Minus, xs4), xs4) [2] term(Cons(Val(int7), xs5)) -> term[Let](Cons(Val(int7), xs5), xs5) [2] term(Cons(LPar, xs6)) -> term[Let](Cons(LPar, xs6), factor[Ite][True][Let](Cons(LPar, xs6), expr(Cons(LPar, xs6)))) [2] term(xs) -> term[Let](xs, Nil) [1] parsexp(xs) -> expr(xs) [1] expr(xs) -> expr[Let](xs, term[Let](xs, factor(xs))) [2] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) [0] factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) [0] factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) [0] factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) [0] factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) [0] factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) [0] factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) [0] term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) [0] expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) [0] term[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(null_head, RPar)), xs, Nil) [0] expr[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](True, x, xs) -> True [0] and(v0, v1) -> null_and [0] !EQ(v0, v1) -> null_!EQ [0] factor[Ite][True][Let](v0, v1) -> Nil [0] term[Let](v0, v1) -> Nil [0] expr[Let](v0, v1) -> Nil [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] factor(v0) -> Nil [0] head(v0) -> null_head [0] member(v0, v1) -> null_member [0] eqAlph(v0, v1) -> null_eqAlph [0] The TRS has the following type information: head :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Cons :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] factor :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] RPar :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Div :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Mul :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Plus :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Minus :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Val :: S:0 -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head LPar :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head factor[Ite][True][Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] member :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph member[Ite][True][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph eqAlph :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph Nil :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] False :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph atom :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] True :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph !EQ :: S:0 -> S:0 -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph notEmpty :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph term :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] term[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] parsexp :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] and :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph S :: S:0 -> S:0 0 :: S:0 factor[Ite][True][Let][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] term[Let][Ite][False][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] expr[Let][Ite][False][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite] null_and :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_!EQ :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_member[Ite][True][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_head :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head null_member :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph null_eqAlph :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_eqAlph Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: RPar => 5 Div => 0 Mul => 3 Plus => 4 Minus => 2 LPar => 1 Nil => 0 False => 1 True => 2 0 => 0 null_and => 0 null_!EQ => 0 null_member[Ite][True][Ite] => 0 null_head => 0 null_member => 0 null_eqAlph => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z = 1, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, int2 >= 0, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, int2 >= 0, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: int >= 0, z = 1 + int, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, int >= 0, z = 1 + int eqAlph(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqAlph(z, z') -{ 1 }-> !EQ(int2, int) :|: int2 >= 0, z' = 1 + int2, int >= 0, z = 1 + int expr(z) -{ 2 }-> expr[Let](xs, term[Let](xs, factor(xs))) :|: xs >= 0, z = xs expr[Let](z, z') -{ 0 }-> 0 :|: xs >= 0, z = xs, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + xs' + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, xs' >= 0, x >= 0, z = xs' factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 5 + xs factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 0 + xs factor(z) -{ 1 }-> xs :|: z = 1 + 3 + xs, xs >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + 4 + xs, xs >= 0 factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 2 + xs factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + xs, expr[Let](1 + 1 + xs, term(1 + 1 + xs))) :|: xs >= 0, z = 1 + 1 + xs factor(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + xs + 0 :|: xs >= 0, z = xs, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + xs' + (1 + 5 + xs) :|: xs >= 0, xs' >= 0, z' = 1 + 5 + xs, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 4 + xs) :|: xs >= 0, xs' >= 0, z' = 1 + 4 + xs, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 3 + xs) :|: xs >= 0, z' = 1 + 3 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 2 + xs) :|: xs >= 0, z' = 1 + 2 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 1 + xs) :|: xs >= 0, z' = 1 + 1 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + (1 + int) + xs) :|: xs >= 0, xs' >= 0, z = xs', int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' member(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 member(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, z' = x, x >= 0, z'' = xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(xs) :|: xs >= 0, z = xs term(z) -{ 1 }-> term[Let](xs, 0) :|: xs >= 0, z = xs term(z) -{ 2 }-> term[Let](1 + 5 + xs'', xs'') :|: z = 1 + 5 + xs'', xs'' >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + xs3, xs3) :|: z = 1 + 4 + xs3, xs3 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + xs2, xs2) :|: xs2 >= 0, z = 1 + 3 + xs2 term(z) -{ 2 }-> term[Let](1 + 2 + xs4, xs4) :|: z = 1 + 2 + xs4, xs4 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + xs6, factor[Ite][True][Let](1 + 1 + xs6, expr(1 + 1 + xs6))) :|: xs6 >= 0, z = 1 + 1 + xs6 term(z) -{ 2 }-> term[Let](1 + 0 + xs1, xs1) :|: z = 1 + 0 + xs1, xs1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: xs >= 0, z = xs, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + xs' + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, xs' >= 0, x >= 0, z = xs' ---------------------------------------- (17) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z = 1, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, int2 >= 0, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, int2 >= 0, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: int >= 0, z = 1 + int, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, int >= 0, z = 1 + int eqAlph(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqAlph(z, z') -{ 1 }-> !EQ(int2, int) :|: int2 >= 0, z' = 1 + int2, int >= 0, z = 1 + int expr(z) -{ 2 }-> expr[Let](xs, term[Let](xs, factor(xs))) :|: xs >= 0, z = xs expr[Let](z, z') -{ 0 }-> 0 :|: xs >= 0, z = xs, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + xs' + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, xs' >= 0, x >= 0, z = xs' factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 5 + xs factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 0 + xs factor(z) -{ 1 }-> xs :|: z = 1 + 3 + xs, xs >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + 4 + xs, xs >= 0 factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 2 + xs factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + xs, expr[Let](1 + 1 + xs, term(1 + 1 + xs))) :|: xs >= 0, z = 1 + 1 + xs factor(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + xs + 0 :|: xs >= 0, z = xs, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + xs' + (1 + 5 + xs) :|: xs >= 0, xs' >= 0, z' = 1 + 5 + xs, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 4 + xs) :|: xs >= 0, xs' >= 0, z' = 1 + 4 + xs, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 3 + xs) :|: xs >= 0, z' = 1 + 3 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 2 + xs) :|: xs >= 0, z' = 1 + 2 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 1 + xs) :|: xs >= 0, z' = 1 + 1 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + (1 + int) + xs) :|: xs >= 0, xs' >= 0, z = xs', int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' member(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 member(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, z' = x, x >= 0, z'' = xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(xs) :|: xs >= 0, z = xs term(z) -{ 1 }-> term[Let](xs, 0) :|: xs >= 0, z = xs term(z) -{ 2 }-> term[Let](1 + 5 + xs'', xs'') :|: z = 1 + 5 + xs'', xs'' >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + xs3, xs3) :|: z = 1 + 4 + xs3, xs3 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + xs2, xs2) :|: xs2 >= 0, z = 1 + 3 + xs2 term(z) -{ 2 }-> term[Let](1 + 2 + xs4, xs4) :|: z = 1 + 2 + xs4, xs4 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + xs6, factor[Ite][True][Let](1 + 1 + xs6, expr(1 + 1 + xs6))) :|: xs6 >= 0, z = 1 + 1 + xs6 term(z) -{ 2 }-> term[Let](1 + 0 + xs1, xs1) :|: z = 1 + 0 + xs1, xs1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: xs >= 0, z = xs, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + xs' + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, xs' >= 0, x >= 0, z = xs' ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { notEmpty } { !EQ } { and } { atom } { head } { eqAlph } { member, member[Ite][True][Ite] } { factor[Ite][True][Let] } { expr[Let] } { term[Let] } { expr, factor, term } { parsexp } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {notEmpty}, {!EQ}, {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {notEmpty}, {!EQ}, {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: notEmpty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {notEmpty}, {!EQ}, {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: ?, size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: notEmpty 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: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {!EQ}, {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {!EQ}, {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: !EQ after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {!EQ}, {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: ?, size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: !EQ after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqAlph(z, z') -{ 1 }-> !EQ(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {and}, {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: ?, size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: atom after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {atom}, {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: ?, size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: atom after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {head}, {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: ?, size: O(n^1) [z] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eqAlph after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {eqAlph}, {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: ?, size: O(1) [2] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: eqAlph after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(0, 5)) + z + 0 :|: z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, z), z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 2 }-> member[Ite][True][Ite](s'', z, 1 + x + xs) :|: s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: member after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 Computed SIZE bound using CoFloCo for: member[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 2 }-> member[Ite][True][Ite](s'', z, 1 + x + xs) :|: s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: ?, size: O(1) [2] member[Ite][True][Ite]: runtime: ?, size: O(1) [2] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: member after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z' Computed RUNTIME bound using CoFloCo for: member[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 2*z'' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 2 }-> member[Ite][True][Ite](s'', z, 1 + x + xs) :|: s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + z + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] ---------------------------------------- (65) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: factor[Ite][True][Let] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z + z' ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {factor[Ite][True][Let]}, {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: ?, size: O(n^1) [3 + z + z'] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: factor[Ite][True][Let] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] ---------------------------------------- (71) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: expr[Let] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z + z' ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr[Let]}, {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: ?, size: O(n^1) [3 + z + z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: expr[Let] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 20 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] ---------------------------------------- (77) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: term[Let] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z + z' ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {term[Let]}, {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] term[Let]: runtime: ?, size: O(n^1) [3 + z + z'] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: term[Let] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 14 ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 1 }-> term[Let](z, 0) :|: z >= 0 term(z) -{ 2 }-> term[Let](1 + 5 + (z - 6), z - 6) :|: z - 6 >= 0 term(z) -{ 2 }-> term[Let](1 + 4 + (z - 5), z - 5) :|: z - 5 >= 0 term(z) -{ 2 }-> term[Let](1 + 3 + (z - 4), z - 4) :|: z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 2 + (z - 3), z - 3) :|: z - 3 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term(z) -{ 2 }-> term[Let](1 + 0 + (z - 1), z - 1) :|: z - 1 >= 0 term(z) -{ 2 }-> term[Let](1 + (1 + int7) + xs5, xs5) :|: int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] term[Let]: runtime: O(1) [14], size: O(n^1) [3 + z + z'] ---------------------------------------- (83) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 16 }-> s10 :|: s10 >= 0, s10 <= 1 + 4 + (z - 5) + (z - 5) + 3, z - 5 >= 0 term(z) -{ 16 }-> s11 :|: s11 >= 0, s11 <= 1 + 2 + (z - 3) + (z - 3) + 3, z - 3 >= 0 term(z) -{ 16 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + int7) + xs5 + xs5 + 3, int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term(z) -{ 15 }-> s13 :|: s13 >= 0, s13 <= z + 0 + 3, z >= 0 term(z) -{ 16 }-> s7 :|: s7 >= 0, s7 <= 1 + 5 + (z - 6) + (z - 6) + 3, z - 6 >= 0 term(z) -{ 16 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (z - 1) + (z - 1) + 3, z - 1 >= 0 term(z) -{ 16 }-> s9 :|: s9 >= 0, s9 <= 1 + 3 + (z - 4) + (z - 4) + 3, z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] term[Let]: runtime: O(1) [14], size: O(n^1) [3 + z + z'] ---------------------------------------- (85) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: expr after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: factor after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: term after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 16 }-> s10 :|: s10 >= 0, s10 <= 1 + 4 + (z - 5) + (z - 5) + 3, z - 5 >= 0 term(z) -{ 16 }-> s11 :|: s11 >= 0, s11 <= 1 + 2 + (z - 3) + (z - 3) + 3, z - 3 >= 0 term(z) -{ 16 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + int7) + xs5 + xs5 + 3, int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term(z) -{ 15 }-> s13 :|: s13 >= 0, s13 <= z + 0 + 3, z >= 0 term(z) -{ 16 }-> s7 :|: s7 >= 0, s7 <= 1 + 5 + (z - 6) + (z - 6) + 3, z - 6 >= 0 term(z) -{ 16 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (z - 1) + (z - 1) + 3, z - 1 >= 0 term(z) -{ 16 }-> s9 :|: s9 >= 0, s9 <= 1 + 3 + (z - 4) + (z - 4) + 3, z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] term[Let]: runtime: O(1) [14], size: O(n^1) [3 + z + z'] expr: runtime: ?, size: INF factor: runtime: ?, size: INF term: runtime: ?, size: INF ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: expr after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 eqAlph(z, z') -{ 1 }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z - 1 >= 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z - 1 >= 0 eqAlph(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr(z) -{ 2 }-> expr[Let](z, term[Let](z, factor(z))) :|: z >= 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 expr[Let](z, z') -{ 20 }-> 1 + s5 + z + (1 + x + xs) :|: s5 >= 0, s5 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 2 }-> factor[Ite][True][Let](1 + 1 + (z - 2), expr[Let](1 + 1 + (z - 2), term(1 + 1 + (z - 2)))) :|: z - 2 >= 0 factor(z) -{ 0 }-> 0 :|: z >= 0 factor(z) -{ 1 }-> z - 6 :|: z - 6 >= 0 factor(z) -{ 1 }-> z - 5 :|: z - 5 >= 0 factor(z) -{ 1 }-> z - 4 :|: z - 4 >= 0 factor(z) -{ 1 }-> z - 3 :|: z - 3 >= 0 factor(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 factor[Ite][True][Let](z, z') -{ 1 }-> 1 + s2 + z + 0 :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z >= 0, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + z + (1 + 5 + (z' - 6)) :|: z' - 6 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 4 + (z' - 5)) :|: z' - 5 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 3 + (z' - 4)) :|: z' - 4 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 2 + (z' - 3)) :|: z' - 3 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + z + (1 + (1 + int) + xs) :|: xs >= 0, z >= 0, int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 member(z, z') -{ 6 + 2*x + 2*xs }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 4 + 2*xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 parsexp(z) -{ 1 }-> expr(z) :|: z >= 0 term(z) -{ 16 }-> s10 :|: s10 >= 0, s10 <= 1 + 4 + (z - 5) + (z - 5) + 3, z - 5 >= 0 term(z) -{ 16 }-> s11 :|: s11 >= 0, s11 <= 1 + 2 + (z - 3) + (z - 3) + 3, z - 3 >= 0 term(z) -{ 16 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + int7) + xs5 + xs5 + 3, int7 >= 0, z = 1 + (1 + int7) + xs5, xs5 >= 0 term(z) -{ 15 }-> s13 :|: s13 >= 0, s13 <= z + 0 + 3, z >= 0 term(z) -{ 16 }-> s7 :|: s7 >= 0, s7 <= 1 + 5 + (z - 6) + (z - 6) + 3, z - 6 >= 0 term(z) -{ 16 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (z - 1) + (z - 1) + 3, z - 1 >= 0 term(z) -{ 16 }-> s9 :|: s9 >= 0, s9 <= 1 + 3 + (z - 4) + (z - 4) + 3, z - 4 >= 0 term(z) -{ 2 }-> term[Let](1 + 1 + (z - 2), factor[Ite][True][Let](1 + 1 + (z - 2), expr(1 + 1 + (z - 2)))) :|: z - 2 >= 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 term[Let](z, z') -{ 14 }-> 1 + s4 + z + (1 + x + xs) :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 Function symbols to be analyzed: {expr,factor,term}, {parsexp} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] atom: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] eqAlph: runtime: O(1) [1], size: O(1) [2] member: runtime: O(n^1) [4 + 2*z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [2 + 2*z''], size: O(1) [2] factor[Ite][True][Let]: runtime: O(1) [1], size: O(n^1) [3 + z + z'] expr[Let]: runtime: O(1) [20], size: O(n^1) [3 + z + z'] term[Let]: runtime: O(1) [14], size: O(n^1) [3 + z + z'] expr: runtime: INF, size: INF factor: runtime: ?, size: INF term: runtime: ?, size: INF ---------------------------------------- (89) 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: and(v0, v1) -> null_and [0] !EQ(v0, v1) -> null_!EQ [0] factor[Ite][True][Let](v0, v1) -> null_factor[Ite][True][Let] [0] term[Let](v0, v1) -> null_term[Let] [0] expr[Let](v0, v1) -> null_expr[Let] [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] head(v0) -> null_head [0] factor(v0) -> null_factor [0] member(v0, v1) -> null_member [0] atom(v0) -> null_atom [0] notEmpty(v0) -> null_notEmpty [0] eqAlph(v0, v1) -> null_eqAlph [0] And the following fresh constants: null_and, null_!EQ, null_factor[Ite][True][Let], null_term[Let], null_expr[Let], null_member[Ite][True][Ite], null_head, null_factor, null_member, null_atom, null_notEmpty, null_eqAlph ---------------------------------------- (90) 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: head(Cons(x, xs)) -> x [1] factor(Cons(RPar, xs)) -> xs [1] factor(Cons(Div, xs)) -> xs [1] factor(Cons(Mul, xs)) -> xs [1] factor(Cons(Plus, xs)) -> xs [1] factor(Cons(Minus, xs)) -> xs [1] factor(Cons(Val(int), xs)) -> xs [1] factor(Cons(LPar, xs)) -> factor[Ite][True][Let](Cons(LPar, xs), expr(Cons(LPar, xs))) [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](eqAlph(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] atom(Cons(x, xs)) -> xs [1] atom(Nil) -> Nil [1] eqAlph(RPar, RPar) -> True [1] eqAlph(RPar, LPar) -> False [1] eqAlph(RPar, Div) -> False [1] eqAlph(RPar, Mul) -> False [1] eqAlph(RPar, Plus) -> False [1] eqAlph(RPar, Minus) -> False [1] eqAlph(RPar, Val(int2)) -> False [1] eqAlph(LPar, RPar) -> False [1] eqAlph(LPar, LPar) -> True [1] eqAlph(LPar, Div) -> False [1] eqAlph(LPar, Mul) -> False [1] eqAlph(LPar, Plus) -> False [1] eqAlph(LPar, Minus) -> False [1] eqAlph(LPar, Val(int2)) -> False [1] eqAlph(Div, RPar) -> False [1] eqAlph(Div, LPar) -> False [1] eqAlph(Div, Div) -> True [1] eqAlph(Div, Mul) -> False [1] eqAlph(Div, Plus) -> False [1] eqAlph(Div, Minus) -> False [1] eqAlph(Div, Val(int2)) -> False [1] eqAlph(Mul, RPar) -> False [1] eqAlph(Mul, LPar) -> False [1] eqAlph(Mul, Div) -> False [1] eqAlph(Mul, Mul) -> True [1] eqAlph(Mul, Plus) -> False [1] eqAlph(Mul, Minus) -> False [1] eqAlph(Mul, Val(int2)) -> False [1] eqAlph(Plus, RPar) -> False [1] eqAlph(Plus, LPar) -> False [1] eqAlph(Plus, Div) -> False [1] eqAlph(Plus, Mul) -> False [1] eqAlph(Plus, Plus) -> True [1] eqAlph(Plus, Minus) -> False [1] eqAlph(Plus, Val(int2)) -> False [1] eqAlph(Minus, RPar) -> False [1] eqAlph(Minus, LPar) -> False [1] eqAlph(Minus, Div) -> False [1] eqAlph(Minus, Mul) -> False [1] eqAlph(Minus, Plus) -> False [1] eqAlph(Minus, Minus) -> True [1] eqAlph(Minus, Val(int2)) -> False [1] eqAlph(Val(int), RPar) -> False [1] eqAlph(Val(int), LPar) -> False [1] eqAlph(Val(int), Div) -> False [1] eqAlph(Val(int), Mul) -> False [1] eqAlph(Val(int), Plus) -> False [1] eqAlph(Val(int), Minus) -> False [1] eqAlph(Val(int), Val(int2)) -> !EQ(int2, int) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] term(xs) -> term[Let](xs, factor(xs)) [1] parsexp(xs) -> expr(xs) [1] expr(xs) -> expr[Let](xs, term(xs)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] factor[Ite][True][Let](xs', Cons(RPar, xs)) -> factor[Ite][True][Let][Ite](True, xs', Cons(RPar, xs)) [0] factor[Ite][True][Let](xs', Cons(LPar, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(LPar, xs)) [0] factor[Ite][True][Let](xs', Cons(Div, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Div, xs)) [0] factor[Ite][True][Let](xs', Cons(Mul, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Mul, xs)) [0] factor[Ite][True][Let](xs', Cons(Plus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Plus, xs)) [0] factor[Ite][True][Let](xs', Cons(Minus, xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Minus, xs)) [0] factor[Ite][True][Let](xs', Cons(Val(int), xs)) -> factor[Ite][True][Let][Ite](False, xs', Cons(Val(int), xs)) [0] term[Let](xs', Cons(x, xs)) -> term[Let][Ite][False][Ite](member(x, Cons(Mul, Cons(Div, Nil))), xs', Cons(x, xs)) [0] expr[Let](xs', Cons(x, xs)) -> expr[Let][Ite][False][Ite](member(x, Cons(Plus, Cons(Minus, Nil))), xs', Cons(x, xs)) [0] term[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] factor[Ite][True][Let](xs, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), xs, Nil) [0] expr[Let](xs, Nil) -> Nil [0] member[Ite][True][Ite](True, x, xs) -> True [0] and(v0, v1) -> null_and [0] !EQ(v0, v1) -> null_!EQ [0] factor[Ite][True][Let](v0, v1) -> null_factor[Ite][True][Let] [0] term[Let](v0, v1) -> null_term[Let] [0] expr[Let](v0, v1) -> null_expr[Let] [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] head(v0) -> null_head [0] factor(v0) -> null_factor [0] member(v0, v1) -> null_member [0] atom(v0) -> null_atom [0] notEmpty(v0) -> null_notEmpty [0] eqAlph(v0, v1) -> null_eqAlph [0] The TRS has the following type information: head :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Cons :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom factor :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom RPar :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Div :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Mul :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Plus :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Minus :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head Val :: S:0 -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head LPar :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head factor[Ite][True][Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom expr :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom member :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph member[Ite][True][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph eqAlph :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> RPar:Div:Mul:Plus:Minus:Val:LPar:null_head -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph Nil :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom False :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph atom :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom True :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph !EQ :: S:0 -> S:0 -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph notEmpty :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph term :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom term[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom parsexp :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom expr[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom and :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph -> False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph S :: S:0 -> S:0 0 :: S:0 factor[Ite][True][Let][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom term[Let][Ite][False][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom expr[Let][Ite][False][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom -> Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom null_and :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph null_!EQ :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph null_factor[Ite][True][Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom null_term[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom null_expr[Let] :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom null_member[Ite][True][Ite] :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph null_head :: RPar:Div:Mul:Plus:Minus:Val:LPar:null_head null_factor :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom null_member :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph null_atom :: Cons:Nil:factor[Ite][True][Let][Ite]:term[Let][Ite][False][Ite]:expr[Let][Ite][False][Ite]:null_factor[Ite][True][Let]:null_term[Let]:null_expr[Let]:null_factor:null_atom null_notEmpty :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph null_eqAlph :: False:True:null_and:null_!EQ:null_member[Ite][True][Ite]:null_member:null_notEmpty:null_eqAlph Rewrite Strategy: INNERMOST ---------------------------------------- (91) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: RPar => 5 Div => 0 Mul => 3 Plus => 4 Minus => 2 LPar => 1 Nil => 0 False => 1 True => 2 0 => 0 null_and => 0 null_!EQ => 0 null_factor[Ite][True][Let] => 0 null_term[Let] => 0 null_expr[Let] => 0 null_member[Ite][True][Ite] => 0 null_head => 0 null_factor => 0 null_member => 0 null_atom => 0 null_notEmpty => 0 null_eqAlph => 0 ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 atom(z) -{ 1 }-> xs :|: z = 1 + x + xs, xs >= 0, x >= 0 atom(z) -{ 1 }-> 0 :|: z = 0 atom(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 5, z' = 5 eqAlph(z, z') -{ 1 }-> 2 :|: z = 1, z' = 1 eqAlph(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqAlph(z, z') -{ 1 }-> 2 :|: z = 3, z' = 3 eqAlph(z, z') -{ 1 }-> 2 :|: z' = 4, z = 4 eqAlph(z, z') -{ 1 }-> 2 :|: z = 2, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 5, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 1, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 1 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z = 1, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 0, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, z = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, z' = 2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 3, int2 >= 0, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 4, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: int2 >= 0, z' = 1 + int2, z = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 5 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 3 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, z' = 4 eqAlph(z, z') -{ 1 }-> 1 :|: z = 2, int2 >= 0, z' = 1 + int2 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 5, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 1, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: int >= 0, z = 1 + int, z' = 0 eqAlph(z, z') -{ 1 }-> 1 :|: z' = 3, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 4, int >= 0, z = 1 + int eqAlph(z, z') -{ 1 }-> 1 :|: z' = 2, int >= 0, z = 1 + int eqAlph(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqAlph(z, z') -{ 1 }-> !EQ(int2, int) :|: int2 >= 0, z' = 1 + int2, int >= 0, z = 1 + int expr(z) -{ 1 }-> expr[Let](xs, term(xs)) :|: xs >= 0, z = xs expr[Let](z, z') -{ 0 }-> 0 :|: xs >= 0, z = xs, z' = 0 expr[Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 expr[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 4 + (1 + 2 + 0)) + xs' + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, xs' >= 0, x >= 0, z = xs' factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 5 + xs factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 0 + xs factor(z) -{ 1 }-> xs :|: z = 1 + 3 + xs, xs >= 0 factor(z) -{ 1 }-> xs :|: z = 1 + 4 + xs, xs >= 0 factor(z) -{ 1 }-> xs :|: xs >= 0, z = 1 + 2 + xs factor(z) -{ 1 }-> xs :|: z = 1 + (1 + int) + xs, xs >= 0, int >= 0 factor(z) -{ 1 }-> factor[Ite][True][Let](1 + 1 + xs, expr(1 + 1 + xs)) :|: xs >= 0, z = 1 + 1 + xs factor(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 factor[Ite][True][Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + and(1, eqAlph(head(0), 5)) + xs + 0 :|: xs >= 0, z = xs, z' = 0 factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 2 + xs' + (1 + 5 + xs) :|: xs >= 0, xs' >= 0, z' = 1 + 5 + xs, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 4 + xs) :|: xs >= 0, xs' >= 0, z' = 1 + 4 + xs, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 3 + xs) :|: xs >= 0, z' = 1 + 3 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 2 + xs) :|: xs >= 0, z' = 1 + 2 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 1 + xs) :|: xs >= 0, z' = 1 + 1 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, xs' >= 0, z = xs' factor[Ite][True][Let](z, z') -{ 0 }-> 1 + 1 + xs' + (1 + (1 + int) + xs) :|: xs >= 0, xs' >= 0, z = xs', int >= 0, z' = 1 + (1 + int) + xs head(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 member(z, z') -{ 1 }-> member[Ite][True][Ite](eqAlph(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' member(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 member(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, z' = x, x >= 0, z'' = xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 notEmpty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 parsexp(z) -{ 1 }-> expr(xs) :|: xs >= 0, z = xs term(z) -{ 1 }-> term[Let](xs, factor(xs)) :|: xs >= 0, z = xs term[Let](z, z') -{ 0 }-> 0 :|: xs >= 0, z = xs, z' = 0 term[Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 term[Let](z, z') -{ 0 }-> 1 + member(x, 1 + 3 + (1 + 0 + 0)) + xs' + (1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, xs' >= 0, x >= 0, z = xs' Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (93) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True head(Cons(z0, z1)) -> z0 factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False atom(Cons(z0, z1)) -> z1 atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False term(z0) -> term[Let](z0, factor(z0)) parsexp(z0) -> expr(z0) expr(z0) -> expr[Let](z0, term(z0)) Tuples: AND(False, False) -> c AND(True, False) -> c1 AND(False, True) -> c2 AND(True, True) -> c3 !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) !EQ'(0, S(z0)) -> c5 !EQ'(S(z0), 0) -> c6 !EQ'(0, 0) -> c7 FACTOR[ITE][TRUE][LET](z0, Cons(RPar, z1)) -> c8 FACTOR[ITE][TRUE][LET](z0, Cons(LPar, z1)) -> c9 FACTOR[ITE][TRUE][LET](z0, Cons(Div, z1)) -> c10 FACTOR[ITE][TRUE][LET](z0, Cons(Mul, z1)) -> c11 FACTOR[ITE][TRUE][LET](z0, Cons(Plus, z1)) -> c12 FACTOR[ITE][TRUE][LET](z0, Cons(Minus, z1)) -> c13 FACTOR[ITE][TRUE][LET](z0, Cons(Val(z1), z2)) -> c14 FACTOR[ITE][TRUE][LET](z0, Nil) -> c15(AND(False, eqAlph(head(Nil), RPar)), EQALPH(head(Nil), RPar), HEAD(Nil)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) TERM[LET](z0, Nil) -> c17 EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Nil) -> c19 MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER[ITE][TRUE][ITE](True, z0, z1) -> c21 HEAD(Cons(z0, z1)) -> c22 FACTOR(Cons(RPar, z0)) -> c23 FACTOR(Cons(Div, z0)) -> c24 FACTOR(Cons(Mul, z0)) -> c25 FACTOR(Cons(Plus, z0)) -> c26 FACTOR(Cons(Minus, z0)) -> c27 FACTOR(Cons(Val(z0), z1)) -> c28 FACTOR(Cons(LPar, z0)) -> c29(FACTOR[ITE][TRUE][LET](Cons(LPar, z0), expr(Cons(LPar, z0))), EXPR(Cons(LPar, z0))) MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 ATOM(Cons(z0, z1)) -> c32 ATOM(Nil) -> c33 EQALPH(RPar, RPar) -> c34 EQALPH(RPar, LPar) -> c35 EQALPH(RPar, Div) -> c36 EQALPH(RPar, Mul) -> c37 EQALPH(RPar, Plus) -> c38 EQALPH(RPar, Minus) -> c39 EQALPH(RPar, Val(z0)) -> c40 EQALPH(LPar, RPar) -> c41 EQALPH(LPar, LPar) -> c42 EQALPH(LPar, Div) -> c43 EQALPH(LPar, Mul) -> c44 EQALPH(LPar, Plus) -> c45 EQALPH(LPar, Minus) -> c46 EQALPH(LPar, Val(z0)) -> c47 EQALPH(Div, RPar) -> c48 EQALPH(Div, LPar) -> c49 EQALPH(Div, Div) -> c50 EQALPH(Div, Mul) -> c51 EQALPH(Div, Plus) -> c52 EQALPH(Div, Minus) -> c53 EQALPH(Div, Val(z0)) -> c54 EQALPH(Mul, RPar) -> c55 EQALPH(Mul, LPar) -> c56 EQALPH(Mul, Div) -> c57 EQALPH(Mul, Mul) -> c58 EQALPH(Mul, Plus) -> c59 EQALPH(Mul, Minus) -> c60 EQALPH(Mul, Val(z0)) -> c61 EQALPH(Plus, RPar) -> c62 EQALPH(Plus, LPar) -> c63 EQALPH(Plus, Div) -> c64 EQALPH(Plus, Mul) -> c65 EQALPH(Plus, Plus) -> c66 EQALPH(Plus, Minus) -> c67 EQALPH(Plus, Val(z0)) -> c68 EQALPH(Minus, RPar) -> c69 EQALPH(Minus, LPar) -> c70 EQALPH(Minus, Div) -> c71 EQALPH(Minus, Mul) -> c72 EQALPH(Minus, Plus) -> c73 EQALPH(Minus, Minus) -> c74 EQALPH(Minus, Val(z0)) -> c75 EQALPH(Val(z0), RPar) -> c76 EQALPH(Val(z0), LPar) -> c77 EQALPH(Val(z0), Div) -> c78 EQALPH(Val(z0), Mul) -> c79 EQALPH(Val(z0), Plus) -> c80 EQALPH(Val(z0), Minus) -> c81 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) NOTEMPTY(Cons(z0, z1)) -> c83 NOTEMPTY(Nil) -> c84 TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) PARSEXP(z0) -> c86(EXPR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) S tuples: HEAD(Cons(z0, z1)) -> c22 FACTOR(Cons(RPar, z0)) -> c23 FACTOR(Cons(Div, z0)) -> c24 FACTOR(Cons(Mul, z0)) -> c25 FACTOR(Cons(Plus, z0)) -> c26 FACTOR(Cons(Minus, z0)) -> c27 FACTOR(Cons(Val(z0), z1)) -> c28 FACTOR(Cons(LPar, z0)) -> c29(FACTOR[ITE][TRUE][LET](Cons(LPar, z0), expr(Cons(LPar, z0))), EXPR(Cons(LPar, z0))) MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 ATOM(Cons(z0, z1)) -> c32 ATOM(Nil) -> c33 EQALPH(RPar, RPar) -> c34 EQALPH(RPar, LPar) -> c35 EQALPH(RPar, Div) -> c36 EQALPH(RPar, Mul) -> c37 EQALPH(RPar, Plus) -> c38 EQALPH(RPar, Minus) -> c39 EQALPH(RPar, Val(z0)) -> c40 EQALPH(LPar, RPar) -> c41 EQALPH(LPar, LPar) -> c42 EQALPH(LPar, Div) -> c43 EQALPH(LPar, Mul) -> c44 EQALPH(LPar, Plus) -> c45 EQALPH(LPar, Minus) -> c46 EQALPH(LPar, Val(z0)) -> c47 EQALPH(Div, RPar) -> c48 EQALPH(Div, LPar) -> c49 EQALPH(Div, Div) -> c50 EQALPH(Div, Mul) -> c51 EQALPH(Div, Plus) -> c52 EQALPH(Div, Minus) -> c53 EQALPH(Div, Val(z0)) -> c54 EQALPH(Mul, RPar) -> c55 EQALPH(Mul, LPar) -> c56 EQALPH(Mul, Div) -> c57 EQALPH(Mul, Mul) -> c58 EQALPH(Mul, Plus) -> c59 EQALPH(Mul, Minus) -> c60 EQALPH(Mul, Val(z0)) -> c61 EQALPH(Plus, RPar) -> c62 EQALPH(Plus, LPar) -> c63 EQALPH(Plus, Div) -> c64 EQALPH(Plus, Mul) -> c65 EQALPH(Plus, Plus) -> c66 EQALPH(Plus, Minus) -> c67 EQALPH(Plus, Val(z0)) -> c68 EQALPH(Minus, RPar) -> c69 EQALPH(Minus, LPar) -> c70 EQALPH(Minus, Div) -> c71 EQALPH(Minus, Mul) -> c72 EQALPH(Minus, Plus) -> c73 EQALPH(Minus, Minus) -> c74 EQALPH(Minus, Val(z0)) -> c75 EQALPH(Val(z0), RPar) -> c76 EQALPH(Val(z0), LPar) -> c77 EQALPH(Val(z0), Div) -> c78 EQALPH(Val(z0), Mul) -> c79 EQALPH(Val(z0), Plus) -> c80 EQALPH(Val(z0), Minus) -> c81 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) NOTEMPTY(Cons(z0, z1)) -> c83 NOTEMPTY(Nil) -> c84 TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) PARSEXP(z0) -> c86(EXPR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) K tuples:none Defined Rule Symbols: head_1, factor_1, member_2, atom_1, eqAlph_2, notEmpty_1, term_1, parsexp_1, expr_1, and_2, !EQ_2, factor[Ite][True][Let]_2, term[Let]_2, expr[Let]_2, member[Ite][True][Ite]_3 Defined Pair Symbols: AND_2, !EQ'_2, FACTOR[ITE][TRUE][LET]_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, HEAD_1, FACTOR_1, MEMBER_2, ATOM_1, EQALPH_2, NOTEMPTY_1, TERM_1, PARSEXP_1, EXPR_1 Compound Symbols: c, c1, c2, c3, c4_1, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15_3, c16_1, c17, c18_1, c19, c20_1, c21, c22, c23, c24, c25, c26, c27, c28, c29_2, c30_2, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82_1, c83, c84, c85_2, c86_1, c87_2 ---------------------------------------- (95) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: PARSEXP(z0) -> c86(EXPR(z0)) Removed 77 trailing nodes: FACTOR(Cons(Plus, z0)) -> c26 EQALPH(Plus, Div) -> c64 EQALPH(LPar, Div) -> c43 EQALPH(LPar, RPar) -> c41 !EQ'(S(z0), 0) -> c6 NOTEMPTY(Cons(z0, z1)) -> c83 EQALPH(RPar, Plus) -> c38 FACTOR(Cons(Val(z0), z1)) -> c28 FACTOR[ITE][TRUE][LET](z0, Cons(Val(z1), z2)) -> c14 EQALPH(RPar, RPar) -> c34 EQALPH(Plus, LPar) -> c63 EQALPH(Val(z0), Mul) -> c79 EQALPH(Plus, Minus) -> c67 FACTOR(Cons(RPar, z0)) -> c23 EQALPH(Div, RPar) -> c48 EQALPH(Plus, Mul) -> c65 FACTOR(Cons(Minus, z0)) -> c27 EQALPH(Val(z0), RPar) -> c76 EQALPH(Mul, Minus) -> c60 EQALPH(Minus, Minus) -> c74 EQALPH(RPar, Minus) -> c39 EQALPH(Mul, Div) -> c57 FACTOR[ITE][TRUE][LET](z0, Cons(Minus, z1)) -> c13 MEMBER[ITE][TRUE][ITE](True, z0, z1) -> c21 !EQ'(0, S(z0)) -> c5 EQALPH(Minus, Plus) -> c73 EQALPH(Val(z0), LPar) -> c77 EQALPH(Mul, LPar) -> c56 EQALPH(Val(z0), Minus) -> c81 EQALPH(Div, Val(z0)) -> c54 FACTOR(Cons(Div, z0)) -> c24 EQALPH(RPar, Val(z0)) -> c40 EQALPH(RPar, Mul) -> c37 EQALPH(Minus, Div) -> c71 EQALPH(RPar, Div) -> c36 EQALPH(Val(z0), Div) -> c78 EQALPH(Div, Div) -> c50 EQALPH(Mul, Mul) -> c58 AND(True, False) -> c1 EQALPH(Mul, Val(z0)) -> c61 EQALPH(Div, LPar) -> c49 EQALPH(LPar, Plus) -> c45 FACTOR(Cons(Mul, z0)) -> c25 ATOM(Nil) -> c33 EQALPH(Div, Minus) -> c53 EQALPH(Minus, Val(z0)) -> c75 EQALPH(Plus, Plus) -> c66 FACTOR[ITE][TRUE][LET](z0, Nil) -> c15(AND(False, eqAlph(head(Nil), RPar)), EQALPH(head(Nil), RPar), HEAD(Nil)) FACTOR[ITE][TRUE][LET](z0, Cons(LPar, z1)) -> c9 EQALPH(Val(z0), Plus) -> c80 EQALPH(RPar, LPar) -> c35 EQALPH(Mul, Plus) -> c59 FACTOR[ITE][TRUE][LET](z0, Cons(Mul, z1)) -> c11 FACTOR[ITE][TRUE][LET](z0, Cons(Plus, z1)) -> c12 FACTOR[ITE][TRUE][LET](z0, Cons(Div, z1)) -> c10 EQALPH(Div, Mul) -> c51 EQALPH(LPar, LPar) -> c42 NOTEMPTY(Nil) -> c84 EQALPH(Minus, LPar) -> c70 FACTOR[ITE][TRUE][LET](z0, Cons(RPar, z1)) -> c8 EQALPH(Div, Plus) -> c52 AND(False, True) -> c2 EQALPH(LPar, Minus) -> c46 EQALPH(Minus, Mul) -> c72 EQALPH(Mul, RPar) -> c55 EQALPH(LPar, Val(z0)) -> c47 EQALPH(Plus, RPar) -> c62 EQALPH(LPar, Mul) -> c44 AND(True, True) -> c3 EQALPH(Minus, RPar) -> c69 HEAD(Cons(z0, z1)) -> c22 EXPR[LET](z0, Nil) -> c19 AND(False, False) -> c ATOM(Cons(z0, z1)) -> c32 EQALPH(Plus, Val(z0)) -> c68 !EQ'(0, 0) -> c7 TERM[LET](z0, Nil) -> c17 ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True head(Cons(z0, z1)) -> z0 factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False atom(Cons(z0, z1)) -> z1 atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False term(z0) -> term[Let](z0, factor(z0)) parsexp(z0) -> expr(z0) expr(z0) -> expr[Let](z0, term(z0)) Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) FACTOR(Cons(LPar, z0)) -> c29(FACTOR[ITE][TRUE][LET](Cons(LPar, z0), expr(Cons(LPar, z0))), EXPR(Cons(LPar, z0))) MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) S tuples: FACTOR(Cons(LPar, z0)) -> c29(FACTOR[ITE][TRUE][LET](Cons(LPar, z0), expr(Cons(LPar, z0))), EXPR(Cons(LPar, z0))) MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) K tuples:none Defined Rule Symbols: head_1, factor_1, member_2, atom_1, eqAlph_2, notEmpty_1, term_1, parsexp_1, expr_1, and_2, !EQ_2, factor[Ite][True][Let]_2, term[Let]_2, expr[Let]_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, FACTOR_1, MEMBER_2, EQALPH_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c29_2, c30_2, c31, c82_1, c85_2, c87_2 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True head(Cons(z0, z1)) -> z0 factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False atom(Cons(z0, z1)) -> z1 atom(Nil) -> Nil eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False term(z0) -> term[Let](z0, factor(z0)) parsexp(z0) -> expr(z0) expr(z0) -> expr[Let](z0, term(z0)) Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) S tuples: MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) K tuples:none Defined Rule Symbols: head_1, factor_1, member_2, atom_1, eqAlph_2, notEmpty_1, term_1, parsexp_1, expr_1, and_2, !EQ_2, factor[Ite][True][Let]_2, term[Let]_2, expr[Let]_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c30_2, c31, c82_1, c85_2, c87_2, c29_1 ---------------------------------------- (99) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True head(Cons(z0, z1)) -> z0 atom(Cons(z0, z1)) -> z1 atom(Nil) -> Nil notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False parsexp(z0) -> expr(z0) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) S tuples: MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) K tuples:none Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c30_2, c31, c82_1, c85_2, c87_2, c29_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MEMBER(z0, Cons(z1, z2)) -> c30(MEMBER[ITE][TRUE][ITE](eqAlph(z1, z0), z0, Cons(z1, z2)), EQALPH(z1, z0)) by MEMBER(RPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, RPar, Cons(RPar, x2)), EQALPH(RPar, RPar)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2)), EQALPH(RPar, LPar)) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2)), EQALPH(RPar, Div)) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2)), EQALPH(RPar, Mul)) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2)), EQALPH(RPar, Plus)) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2)), EQALPH(RPar, Minus)) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2)), EQALPH(RPar, Val(z0))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2)), EQALPH(LPar, RPar)) MEMBER(LPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, LPar, Cons(LPar, x2)), EQALPH(LPar, LPar)) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2)), EQALPH(LPar, Div)) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2)), EQALPH(LPar, Mul)) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2)), EQALPH(LPar, Plus)) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2)), EQALPH(LPar, Minus)) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2)), EQALPH(LPar, Val(z0))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2)), EQALPH(Div, RPar)) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2)), EQALPH(Div, LPar)) MEMBER(Div, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Div, Cons(Div, x2)), EQALPH(Div, Div)) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2)), EQALPH(Div, Mul)) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2)), EQALPH(Div, Plus)) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2)), EQALPH(Div, Minus)) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2)), EQALPH(Div, Val(z0))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2)), EQALPH(Mul, RPar)) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2)), EQALPH(Mul, LPar)) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2)), EQALPH(Mul, Div)) MEMBER(Mul, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Mul, Cons(Mul, x2)), EQALPH(Mul, Mul)) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2)), EQALPH(Mul, Plus)) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2)), EQALPH(Mul, Minus)) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2)), EQALPH(Mul, Val(z0))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2)), EQALPH(Plus, RPar)) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2)), EQALPH(Plus, LPar)) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2)), EQALPH(Plus, Div)) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2)), EQALPH(Plus, Mul)) MEMBER(Plus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Plus, Cons(Plus, x2)), EQALPH(Plus, Plus)) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2)), EQALPH(Plus, Minus)) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2)), EQALPH(Plus, Val(z0))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2)), EQALPH(Minus, RPar)) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2)), EQALPH(Minus, LPar)) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2)), EQALPH(Minus, Div)) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2)), EQALPH(Minus, Mul)) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2)), EQALPH(Minus, Plus)) MEMBER(Minus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Minus, Cons(Minus, x2)), EQALPH(Minus, Minus)) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2)), EQALPH(Minus, Val(z0))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2)), EQALPH(Val(z0), RPar)) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2)), EQALPH(Val(z0), LPar)) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2)), EQALPH(Val(z0), Div)) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2)), EQALPH(Val(z0), Mul)) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2)), EQALPH(Val(z0), Plus)) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2)), EQALPH(Val(z0), Minus)) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(RPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, RPar, Cons(RPar, x2)), EQALPH(RPar, RPar)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2)), EQALPH(RPar, LPar)) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2)), EQALPH(RPar, Div)) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2)), EQALPH(RPar, Mul)) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2)), EQALPH(RPar, Plus)) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2)), EQALPH(RPar, Minus)) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2)), EQALPH(RPar, Val(z0))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2)), EQALPH(LPar, RPar)) MEMBER(LPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, LPar, Cons(LPar, x2)), EQALPH(LPar, LPar)) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2)), EQALPH(LPar, Div)) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2)), EQALPH(LPar, Mul)) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2)), EQALPH(LPar, Plus)) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2)), EQALPH(LPar, Minus)) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2)), EQALPH(LPar, Val(z0))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2)), EQALPH(Div, RPar)) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2)), EQALPH(Div, LPar)) MEMBER(Div, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Div, Cons(Div, x2)), EQALPH(Div, Div)) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2)), EQALPH(Div, Mul)) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2)), EQALPH(Div, Plus)) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2)), EQALPH(Div, Minus)) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2)), EQALPH(Div, Val(z0))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2)), EQALPH(Mul, RPar)) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2)), EQALPH(Mul, LPar)) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2)), EQALPH(Mul, Div)) MEMBER(Mul, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Mul, Cons(Mul, x2)), EQALPH(Mul, Mul)) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2)), EQALPH(Mul, Plus)) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2)), EQALPH(Mul, Minus)) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2)), EQALPH(Mul, Val(z0))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2)), EQALPH(Plus, RPar)) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2)), EQALPH(Plus, LPar)) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2)), EQALPH(Plus, Div)) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2)), EQALPH(Plus, Mul)) MEMBER(Plus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Plus, Cons(Plus, x2)), EQALPH(Plus, Plus)) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2)), EQALPH(Plus, Minus)) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2)), EQALPH(Plus, Val(z0))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2)), EQALPH(Minus, RPar)) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2)), EQALPH(Minus, LPar)) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2)), EQALPH(Minus, Div)) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2)), EQALPH(Minus, Mul)) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2)), EQALPH(Minus, Plus)) MEMBER(Minus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Minus, Cons(Minus, x2)), EQALPH(Minus, Minus)) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2)), EQALPH(Minus, Val(z0))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2)), EQALPH(Val(z0), RPar)) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2)), EQALPH(Val(z0), LPar)) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2)), EQALPH(Val(z0), Div)) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2)), EQALPH(Val(z0), Mul)) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2)), EQALPH(Val(z0), Plus)) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2)), EQALPH(Val(z0), Minus)) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(RPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, RPar, Cons(RPar, x2)), EQALPH(RPar, RPar)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2)), EQALPH(RPar, LPar)) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2)), EQALPH(RPar, Div)) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2)), EQALPH(RPar, Mul)) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2)), EQALPH(RPar, Plus)) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2)), EQALPH(RPar, Minus)) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2)), EQALPH(RPar, Val(z0))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2)), EQALPH(LPar, RPar)) MEMBER(LPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, LPar, Cons(LPar, x2)), EQALPH(LPar, LPar)) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2)), EQALPH(LPar, Div)) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2)), EQALPH(LPar, Mul)) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2)), EQALPH(LPar, Plus)) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2)), EQALPH(LPar, Minus)) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2)), EQALPH(LPar, Val(z0))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2)), EQALPH(Div, RPar)) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2)), EQALPH(Div, LPar)) MEMBER(Div, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Div, Cons(Div, x2)), EQALPH(Div, Div)) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2)), EQALPH(Div, Mul)) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2)), EQALPH(Div, Plus)) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2)), EQALPH(Div, Minus)) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2)), EQALPH(Div, Val(z0))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2)), EQALPH(Mul, RPar)) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2)), EQALPH(Mul, LPar)) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2)), EQALPH(Mul, Div)) MEMBER(Mul, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Mul, Cons(Mul, x2)), EQALPH(Mul, Mul)) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2)), EQALPH(Mul, Plus)) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2)), EQALPH(Mul, Minus)) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2)), EQALPH(Mul, Val(z0))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2)), EQALPH(Plus, RPar)) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2)), EQALPH(Plus, LPar)) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2)), EQALPH(Plus, Div)) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2)), EQALPH(Plus, Mul)) MEMBER(Plus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Plus, Cons(Plus, x2)), EQALPH(Plus, Plus)) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2)), EQALPH(Plus, Minus)) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2)), EQALPH(Plus, Val(z0))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2)), EQALPH(Minus, RPar)) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2)), EQALPH(Minus, LPar)) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2)), EQALPH(Minus, Div)) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2)), EQALPH(Minus, Mul)) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2)), EQALPH(Minus, Plus)) MEMBER(Minus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Minus, Cons(Minus, x2)), EQALPH(Minus, Minus)) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2)), EQALPH(Minus, Val(z0))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2)), EQALPH(Val(z0), RPar)) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2)), EQALPH(Val(z0), LPar)) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2)), EQALPH(Val(z0), Div)) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2)), EQALPH(Val(z0), Mul)) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2)), EQALPH(Val(z0), Plus)) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2)), EQALPH(Val(z0), Minus)) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) K tuples:none Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c85_2, c87_2, c29_1, c30_2 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 54 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) K tuples:none Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c85_2, c87_2, c29_1, c30_2, c30, c30_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(RPar, Cons(RPar, x2)) -> c30 We considered the (Usable) Rules:none And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2 POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = x_1 + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = 0 POL(EXPR[LET](x_1, x_2)) = 0 POL(FACTOR(x_1)) = 0 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = x_2 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_3 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = [1] POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = 0 POL(TERM[LET](x_1, x_2)) = 0 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = [1] + x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = [1] + x_1 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = [1] + x_2 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = x_1 POL(term[Let](x_1, x_2)) = [1] + x_1 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) K tuples: MEMBER(RPar, Cons(RPar, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c85_2, c87_2, c29_1, c30_2, c30, c30_1 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(LPar, Cons(LPar, x2)) -> c30 We considered the (Usable) Rules:none And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2 POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = x_1 + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = 0 POL(EXPR[LET](x_1, x_2)) = 0 POL(FACTOR(x_1)) = 0 POL(False) = 0 POL(LPar) = [1] POL(MEMBER(x_1, x_2)) = x_2 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_3 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = 0 POL(TERM[LET](x_1, x_2)) = 0 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = [1] + x_1 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = [1] + x_1 + x_2 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = x_1 POL(term[Let](x_1, x_2)) = [1] + x_1 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) K tuples: MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(LPar, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c85_2, c87_2, c29_1, c30_2, c30, c30_1 ---------------------------------------- (109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(Minus, Cons(Minus, x2)) -> c30 We considered the (Usable) Rules: term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = 0 POL(!EQ'(x_1, x_2)) = 0 POL(0) = 0 POL(Cons(x_1, x_2)) = x_1 + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = 0 POL(EXPR(x_1)) = 0 POL(EXPR[LET](x_1, x_2)) = [2]x_2 POL(FACTOR(x_1)) = 0 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = [2]x_1*x_2 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = [2]x_2*x_3 POL(Minus) = [1] POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = 0 POL(TERM(x_1)) = 0 POL(TERM[LET](x_1, x_2)) = 0 POL(True) = 0 POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = 0 POL(expr(x_1)) = 0 POL(expr[Let](x_1, x_2)) = [1] + x_1 + x_1^2 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 POL(factor(x_1)) = 0 POL(factor[Ite][True][Let](x_1, x_2)) = [1] POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = [1] + x_2 POL(head(x_1)) = 0 POL(member(x_1, x_2)) = x_1^2 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_2^2 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) K tuples: MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Minus, Cons(Minus, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, TERM_1, EXPR_1, FACTOR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c85_2, c87_2, c29_1, c30_2, c30, c30_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TERM(z0) -> c85(TERM[LET](z0, factor(z0)), FACTOR(z0)) by TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0), FACTOR(Cons(RPar, z0))) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0), FACTOR(Cons(Div, z0))) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0), FACTOR(Cons(Mul, z0))) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0), FACTOR(Cons(Plus, z0))) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0), FACTOR(Cons(Minus, z0))) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1), FACTOR(Cons(Val(z0), z1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0), FACTOR(Cons(RPar, z0))) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0), FACTOR(Cons(Div, z0))) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0), FACTOR(Cons(Mul, z0))) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0), FACTOR(Cons(Plus, z0))) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0), FACTOR(Cons(Minus, z0))) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1), FACTOR(Cons(Val(z0), z1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0), FACTOR(Cons(RPar, z0))) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0), FACTOR(Cons(Div, z0))) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0), FACTOR(Cons(Mul, z0))) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0), FACTOR(Cons(Plus, z0))) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0), FACTOR(Cons(Minus, z0))) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1), FACTOR(Cons(Val(z0), z1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(RPar, Cons(RPar, x2)) -> c30 MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Minus, Cons(Minus, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30, c30_1, c85_2 ---------------------------------------- (113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MEMBER(LPar, Cons(LPar, x2)) -> c30 MEMBER(Minus, Cons(Minus, x2)) -> c30 MEMBER(RPar, Cons(RPar, x2)) -> c30 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0), FACTOR(Cons(RPar, z0))) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0), FACTOR(Cons(Div, z0))) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0), FACTOR(Cons(Mul, z0))) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0), FACTOR(Cons(Plus, z0))) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0), FACTOR(Cons(Minus, z0))) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1), FACTOR(Cons(Val(z0), z1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0), FACTOR(Cons(RPar, z0))) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0), FACTOR(Cons(Div, z0))) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0), FACTOR(Cons(Mul, z0))) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0), FACTOR(Cons(Plus, z0))) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0), FACTOR(Cons(Minus, z0))) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1), FACTOR(Cons(Val(z0), z1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples:none Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2 ---------------------------------------- (115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) K tuples:none Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (117) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) We considered the (Usable) Rules:none And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2 POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = [1] POL(Div) = 0 POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = [1] + x_1 POL(EXPR[LET](x_1, x_2)) = 0 POL(FACTOR(x_1)) = [1] + x_1 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = 0 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = [1] + x_1 POL(TERM[LET](x_1, x_2)) = 0 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = [1] + x_1 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = [1] + x_1 + x_2 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 + x_2 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = x_1 POL(term[Let](x_1, x_2)) = [1] + x_1 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(Mul, Cons(Mul, x2)) -> c30 We considered the (Usable) Rules: factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) term(z0) -> term[Let](z0, factor(z0)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Nil) -> Nil factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2 POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = x_1 + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = x_1 POL(EXPR[LET](x_1, x_2)) = x_2 POL(FACTOR(x_1)) = x_1 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = x_1 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_2 POL(Minus) = 0 POL(Mul) = [1] POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = x_1 POL(TERM[LET](x_1, x_2)) = x_2 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = [1] + x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = 0 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = 0 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(Div, Cons(Div, x2)) -> c30 We considered the (Usable) Rules: factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) term(z0) -> term[Let](z0, factor(z0)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Nil) -> Nil factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2 POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = x_1 + x_2 POL(Div) = [1] POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = x_1 POL(EXPR[LET](x_1, x_2)) = x_2 POL(FACTOR(x_1)) = x_1 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = x_1 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_2 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = x_1 POL(TERM[LET](x_1, x_2)) = x_2 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = 0 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = 0 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(Plus, Cons(Plus, x2)) -> c30 We considered the (Usable) Rules: factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) term(z0) -> term[Let](z0, factor(z0)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Nil) -> Nil factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2 POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = x_1 + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = x_1 POL(EXPR[LET](x_1, x_2)) = x_2 POL(FACTOR(x_1)) = x_1 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = x_1 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_2 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = [1] POL(RPar) = 0 POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = x_1 POL(TERM[LET](x_1, x_2)) = x_2 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = 0 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = 0 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Plus, Cons(Plus, x2)) -> c30 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(z0, Nil) -> c31 We considered the (Usable) Rules: !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Nil) -> Nil !EQ(0, 0) -> True factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = [1] POL(!EQ'(x_1, x_2)) = 0 POL(0) = [1] POL(Cons(x_1, x_2)) = [1] + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = x_1 + x_2 POL(EXPR(x_1)) = x_1 POL(EXPR[LET](x_1, x_2)) = x_2 POL(FACTOR(x_1)) = x_1 POL(False) = [1] POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = [1] POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_1 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = [1] + x_1 POL(TERM(x_1)) = x_1 POL(TERM[LET](x_1, x_2)) = x_2 POL(True) = [1] POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = x_2 POL(expr(x_1)) = [1] + x_1 POL(expr[Let](x_1, x_2)) = [1] + x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(factor(x_1)) = [1] POL(factor[Ite][True][Let](x_1, x_2)) = 0 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = 0 POL(head(x_1)) = [1] + x_1 POL(member(x_1, x_2)) = [1] + x_1 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(z0, Nil) -> c31 Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (127) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) We considered the (Usable) Rules: factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) term(z0) -> term[Let](z0, factor(z0)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Nil) -> Nil factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = 0 POL(!EQ'(x_1, x_2)) = [1] POL(0) = 0 POL(Cons(x_1, x_2)) = [3] + x_2 POL(Div) = [2] POL(EQALPH(x_1, x_2)) = x_1 POL(EXPR(x_1)) = [3]x_1 POL(EXPR[LET](x_1, x_2)) = [3]x_2 POL(FACTOR(x_1)) = [3]x_1 POL(False) = 0 POL(LPar) = [1] POL(MEMBER(x_1, x_2)) = [2] + x_2 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_2 + x_3 POL(Minus) = [1] POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = [1] POL(RPar) = [1] POL(S(x_1)) = x_1 POL(TERM(x_1)) = [3]x_1 POL(TERM[LET](x_1, x_2)) = [3]x_2 POL(True) = 0 POL(Val(x_1)) = [1] POL(and(x_1, x_2)) = [3] + x_1 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = [3] + x_1 POL(expr(x_1)) = 0 POL(expr[Let](x_1, x_2)) = [3] + [3]x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [3] + x_2 POL(factor(x_1)) = [2] POL(factor[Ite][True][Let](x_1, x_2)) = 0 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = 0 POL(head(x_1)) = [3] POL(member(x_1, x_2)) = [3] + [3]x_1 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [3] + [3]x_2 + [3]x_3 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(z0, Nil) -> c31 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) We considered the (Usable) Rules: factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) term(z0) -> term[Let](z0, factor(z0)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) term[Let](z0, Nil) -> Nil factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) And the Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(!EQ(x_1, x_2)) = 0 POL(!EQ'(x_1, x_2)) = 0 POL(0) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(Div) = 0 POL(EQALPH(x_1, x_2)) = [3]x_1 + [3]x_2 POL(EXPR(x_1)) = [3]x_1 POL(EXPR[LET](x_1, x_2)) = [3]x_2 POL(FACTOR(x_1)) = [3]x_1 POL(False) = 0 POL(LPar) = 0 POL(MEMBER(x_1, x_2)) = [1] + x_2 POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_3 POL(Minus) = 0 POL(Mul) = 0 POL(Nil) = 0 POL(Plus) = 0 POL(RPar) = 0 POL(S(x_1)) = x_1 POL(TERM(x_1)) = [3]x_1 POL(TERM[LET](x_1, x_2)) = [3]x_2 POL(True) = 0 POL(Val(x_1)) = 0 POL(and(x_1, x_2)) = [3] + x_1 POL(c16(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c30(x_1)) = x_1 POL(c30(x_1, x_2)) = x_1 + x_2 POL(c31) = 0 POL(c4(x_1)) = x_1 POL(c82(x_1)) = x_1 POL(c85(x_1)) = x_1 POL(c85(x_1, x_2)) = x_1 + x_2 POL(c87(x_1, x_2)) = x_1 + x_2 POL(eqAlph(x_1, x_2)) = [3] + [3]x_1 POL(expr(x_1)) = 0 POL(expr[Let](x_1, x_2)) = [3] + [3]x_1 POL(expr[Let][Ite][False][Ite](x_1, x_2, x_3)) = [3] + x_2 + x_3 POL(factor(x_1)) = [2] POL(factor[Ite][True][Let](x_1, x_2)) = 0 POL(factor[Ite][True][Let][Ite](x_1, x_2, x_3)) = 0 POL(head(x_1)) = [3] POL(member(x_1, x_2)) = [3] + [3]x_1 POL(member[Ite][True][Ite](x_1, x_2, x_3)) = [3] + [3]x_2 + [3]x_3 POL(term(x_1)) = 0 POL(term[Let](x_1, x_2)) = 0 POL(term[Let][Ite][False][Ite](x_1, x_2, x_3)) = 0 ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(z0, Nil) -> c31 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (131) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) S tuples: EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(z0, Nil) -> c31 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, EXPR_1, FACTOR_1, TERM_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c87_2, c29_1, c30_2, c30_1, c30, c85_2, c85_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace EXPR(z0) -> c87(EXPR[LET](z0, term(z0)), TERM(z0)) by EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) MEMBER(z0, Nil) -> c31 EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(Div, Cons(Div, x2)) -> c30 MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(z0, Nil) -> c31 MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, EQALPH_2, FACTOR_1, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c31, c82_1, c29_1, c30_2, c30_1, c30, c85_2, c85_1, c87_2 ---------------------------------------- (135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: MEMBER(Plus, Cons(Plus, x2)) -> c30 MEMBER(Mul, Cons(Mul, x2)) -> c30 MEMBER(z0, Nil) -> c31 MEMBER(Div, Cons(Div, x2)) -> c30 ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_2, c30_1, c85_2, c85_1, c87_2 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) by MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Val(0), Cons(Val(0), x2)), EQALPH(Val(0), Val(0))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](True, Val(0), Cons(Val(0), x2)), EQALPH(Val(0), Val(0))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c85_2, c85_1, c87_2, c30_2 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c85_2, c85_1, c87_2, c30_2 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0)))), FACTOR(Cons(LPar, z0))) by TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c85_1, c87_2, c30_2, c85_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace EXPR(z0) -> c87(EXPR[LET](z0, term[Let](z0, factor(z0))), TERM(z0)) by EXPR(Cons(RPar, z0)) -> c87(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0)), TERM(Cons(RPar, z0))) EXPR(Cons(Div, z0)) -> c87(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0)), TERM(Cons(Div, z0))) EXPR(Cons(Mul, z0)) -> c87(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0)), TERM(Cons(Mul, z0))) EXPR(Cons(Plus, z0)) -> c87(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0)), TERM(Cons(Plus, z0))) EXPR(Cons(Minus, z0)) -> c87(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0)), TERM(Cons(Minus, z0))) EXPR(Cons(Val(z0), z1)) -> c87(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1)), TERM(Cons(Val(z0), z1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c87(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0)), TERM(Cons(RPar, z0))) EXPR(Cons(Div, z0)) -> c87(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0)), TERM(Cons(Div, z0))) EXPR(Cons(Mul, z0)) -> c87(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0)), TERM(Cons(Mul, z0))) EXPR(Cons(Plus, z0)) -> c87(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0)), TERM(Cons(Plus, z0))) EXPR(Cons(Minus, z0)) -> c87(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0)), TERM(Cons(Minus, z0))) EXPR(Cons(Val(z0), z1)) -> c87(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1)), TERM(Cons(Val(z0), z1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c87(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0)), TERM(Cons(RPar, z0))) EXPR(Cons(Div, z0)) -> c87(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0)), TERM(Cons(Div, z0))) EXPR(Cons(Mul, z0)) -> c87(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0)), TERM(Cons(Mul, z0))) EXPR(Cons(Plus, z0)) -> c87(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0)), TERM(Cons(Plus, z0))) EXPR(Cons(Minus, z0)) -> c87(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0)), TERM(Cons(Minus, z0))) EXPR(Cons(Val(z0), z1)) -> c87(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1)), TERM(Cons(Val(z0), z1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c85_1, c30_2, c85_2, c87_2 ---------------------------------------- (145) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(RPar, z0)) -> c(TERM(Cons(RPar, z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Div, z0)) -> c(TERM(Cons(Div, z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Mul, z0)) -> c(TERM(Cons(Mul, z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Plus, z0)) -> c(TERM(Cons(Plus, z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(TERM(Cons(Minus, z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EXPR(Cons(Val(z0), z1)) -> c(TERM(Cons(Val(z0), z1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(RPar, z0)) -> c(TERM(Cons(RPar, z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Div, z0)) -> c(TERM(Cons(Div, z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Mul, z0)) -> c(TERM(Cons(Mul, z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Plus, z0)) -> c(TERM(Cons(Plus, z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(TERM(Cons(Minus, z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EXPR(Cons(Val(z0), z1)) -> c(TERM(Cons(Val(z0), z1))) K tuples: TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(z1), Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z1, z0), Val(z1), Cons(Val(z0), x2)), EQALPH(Val(z0), Val(z1))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c85_1, c30_2, c85_2, c87_2, c_1 ---------------------------------------- (147) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 12 leading nodes: EXPR(Cons(RPar, z0)) -> c(TERM(Cons(RPar, z0))) TERM(Cons(RPar, z0)) -> c85(TERM[LET](Cons(RPar, z0), z0)) EXPR(Cons(Div, z0)) -> c(TERM(Cons(Div, z0))) TERM(Cons(Div, z0)) -> c85(TERM[LET](Cons(Div, z0), z0)) EXPR(Cons(Mul, z0)) -> c(TERM(Cons(Mul, z0))) TERM(Cons(Mul, z0)) -> c85(TERM[LET](Cons(Mul, z0), z0)) EXPR(Cons(Plus, z0)) -> c(TERM(Cons(Plus, z0))) TERM(Cons(Plus, z0)) -> c85(TERM[LET](Cons(Plus, z0), z0)) EXPR(Cons(Minus, z0)) -> c(TERM(Cons(Minus, z0))) TERM(Cons(Minus, z0)) -> c85(TERM[LET](Cons(Minus, z0), z0)) EXPR(Cons(Val(z0), z1)) -> c(TERM(Cons(Val(z0), z1))) TERM(Cons(Val(z0), z1)) -> c85(TERM[LET](Cons(Val(z0), z1), z1)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c30_2, c85_2, c85_1, c87_2, c_1 ---------------------------------------- (149) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, TERM[LET]_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1 Compound Symbols: c4_1, c16_1, c18_1, c20_1, c82_1, c29_1, c30_1, c30_2, c85_2, c85_1, c87_2, c_1 ---------------------------------------- (151) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace TERM[LET](z0, Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) by TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2 Compound Symbols: c4_1, c18_1, c20_1, c82_1, c29_1, c30_1, c30_2, c85_2, c85_1, c87_2, c_1, c16_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace !EQ'(S(z0), S(z1)) -> c4(!EQ'(z0, z1)) by !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2 Compound Symbols: c18_1, c20_1, c82_1, c29_1, c30_1, c30_2, c85_2, c85_1, c87_2, c_1, c16_1, c4_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR[LET](z0, Cons(z1, z2)) -> c18(MEMBER(z1, Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](z0, Cons(RPar, z2)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(RPar, z2)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2 Compound Symbols: c20_1, c82_1, c29_1, c30_1, c30_2, c85_2, c85_1, c87_2, c_1, c16_1, c4_1, c18_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace TERM(Cons(LPar, x0)) -> c85(TERM[LET](Cons(LPar, x0), factor[Ite][True][Let](Cons(LPar, x0), expr[Let](Cons(LPar, x0), term(Cons(LPar, x0))))), FACTOR(Cons(LPar, x0))) by TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(RPar, z2)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2 Compound Symbols: c20_1, c82_1, c29_1, c30_1, c30_2, c85_1, c87_2, c_1, c16_1, c4_1, c18_1, c85_2 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))), TERM(Cons(LPar, z0))) by EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(RPar, z2)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: MEMBER[ITE][TRUE][ITE]_3, EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2 Compound Symbols: c20_1, c82_1, c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2 ---------------------------------------- (161) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c20(MEMBER(z0, z2)) by MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(RPar, z2)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3 Compound Symbols: c82_1, c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1 ---------------------------------------- (163) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace EXPR[LET](z0, Cons(RPar, z2)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: EQALPH_2, FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3 Compound Symbols: c82_1, c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQALPH(Val(z0), Val(z1)) -> c82(!EQ'(z1, z0)) by EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MEMBER(Val(0), Cons(Val(0), x2)) -> c30(EQALPH(Val(0), Val(0))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2)), EQALPH(Val(S(z0)), Val(0))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2)), EQALPH(Val(0), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (171) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace EXPR[LET](z0, Cons(LPar, z2)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](Cons(RPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(RPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (173) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace EXPR[LET](z0, Cons(Div, z2)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](Cons(RPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(RPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(RPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(Val(x0), Cons(Val(x1), x2)) -> c30(EQALPH(Val(x1), Val(x0))) by MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(RPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(RPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR(Cons(RPar, z0)) -> c(EXPR[LET](Cons(RPar, z0), term[Let](Cons(RPar, z0), z0))) by none ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(RPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(RPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (179) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: EXPR[LET](Cons(RPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(RPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(RPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (181) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace EXPR[LET](z0, Cons(Mul, z2)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](Cons(Div, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Div, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (183) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace EXPR[LET](z0, Cons(Minus, z2)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](Cons(Div, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Div, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, EXPR[LET]_2, MEMBER[ITE][TRUE][ITE]_3, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c18_1, c85_2, c87_2, c20_1, c82_1 ---------------------------------------- (185) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace EXPR[LET](z0, Cons(Val(y0), z2)) -> c18(MEMBER(Val(y0), Cons(Plus, Cons(Minus, Nil)))) by EXPR[LET](Cons(Div, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Div, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c85_2, c87_2, c20_1, c18_1, c82_1 ---------------------------------------- (187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR(Cons(Div, z0)) -> c(EXPR[LET](Cons(Div, z0), term[Let](Cons(Div, z0), z0))) by none ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Div, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c85_2, c87_2, c20_1, c18_1, c82_1 ---------------------------------------- (189) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 6 leading nodes: EXPR[LET](Cons(Div, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Div, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c85_2, c87_2, c20_1, c18_1, c82_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR(Cons(Mul, z0)) -> c(EXPR[LET](Cons(Mul, z0), term[Let](Cons(Mul, z0), z0))) by none ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c85_2, c87_2, c20_1, c18_1, c82_1 ---------------------------------------- (193) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 6 leading nodes: EXPR[LET](Cons(Mul, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Mul, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c85_2, c87_2, c20_1, c18_1, c82_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0)))))), FACTOR(Cons(LPar, z0))) by TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR(Cons(Plus, z0)) -> c(EXPR[LET](Cons(Plus, z0), term[Let](Cons(Plus, z0), z0))) by none ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (199) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 6 leading nodes: EXPR[LET](Cons(Plus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Plus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR(Cons(Minus, z0)) -> c(EXPR[LET](Cons(Minus, z0), term[Let](Cons(Minus, z0), z0))) by none ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (203) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 6 leading nodes: EXPR[LET](Cons(Minus, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Minus, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, TERM[LET]_2, !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (205) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EXPR(Cons(Val(z0), z1)) -> c(EXPR[LET](Cons(Val(z0), z1), term[Let](Cons(Val(z0), z1), z1))) by none ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, TERM[LET]_2, !EQ'_2, EXPR_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (207) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 6 leading nodes: EXPR[LET](Cons(Val(x0), x1), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(Val(x0), x1), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, TERM[LET]_2, !EQ'_2, EXPR_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c16_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2 ---------------------------------------- (209) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TERM[LET](Cons(LPar, x0), Cons(z1, z2)) -> c16(MEMBER(z1, Cons(Mul, Cons(Div, Nil)))) by TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, !EQ'_2, EXPR_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c4_1, c87_2, c20_1, c18_1, c82_1, c85_2, c16_1 ---------------------------------------- (211) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace !EQ'(S(S(y0)), S(S(y1))) -> c4(!EQ'(S(y0), S(y1))) by !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c87_2, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1 ---------------------------------------- (213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x0)) -> c20(MEMBER(LPar, x0)) by MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c87_2, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1 ---------------------------------------- (215) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(LPar, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, x2))) by MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) K tuples: MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, EXPR_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c87_2, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1 ---------------------------------------- (217) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term(Cons(LPar, z0)))))), TERM(Cons(LPar, z0))) by EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (219) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x0)) -> c20(MEMBER(Div, x0)) by MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (221) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(Div, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, x2))) by MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (223) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x0)) -> c20(MEMBER(Mul, x0)) by MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (225) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(Mul, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, x2))) by MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (227) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x0)) -> c20(MEMBER(Plus, x0)) by MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (229) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(Plus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, x2))) by MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (231) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x0)) -> c20(MEMBER(Minus, x0)) by MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (233) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(Minus, Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, x2))) by MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (235) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(RPar, x1)) -> c20(MEMBER(Val(x0), x1)) by MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1))) -> c20(MEMBER(Val(z0), Cons(RPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1))) -> c20(MEMBER(Val(z0), Cons(LPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1))) -> c20(MEMBER(Val(z0), Cons(Div, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1))) -> c20(MEMBER(Val(z0), Cons(Mul, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1))) -> c20(MEMBER(Val(z0), Cons(Plus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1))) -> c20(MEMBER(Val(z0), Cons(Minus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c20(MEMBER(Val(S(y0)), Cons(Val(S(y1)), y2))) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c20(MEMBER(Val(0), Cons(Val(S(y0)), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c20(MEMBER(Val(S(y0)), Cons(Val(0), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c20(MEMBER(Val(S(S(y0))), Cons(Val(S(S(y1))), y2))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1))) -> c20(MEMBER(Val(z0), Cons(RPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1))) -> c20(MEMBER(Val(z0), Cons(LPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1))) -> c20(MEMBER(Val(z0), Cons(Div, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1))) -> c20(MEMBER(Val(z0), Cons(Mul, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1))) -> c20(MEMBER(Val(z0), Cons(Plus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1))) -> c20(MEMBER(Val(z0), Cons(Minus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c20(MEMBER(Val(S(y0)), Cons(Val(S(y1)), y2))) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c20(MEMBER(Val(0), Cons(Val(S(y0)), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c20(MEMBER(Val(S(y0)), Cons(Val(0), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c20(MEMBER(Val(S(S(y0))), Cons(Val(S(S(y1))), y2))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (237) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(Val(z0), Cons(RPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, x2))) by MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1))) -> c20(MEMBER(Val(z0), Cons(RPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1))) -> c20(MEMBER(Val(z0), Cons(LPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1))) -> c20(MEMBER(Val(z0), Cons(Div, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1))) -> c20(MEMBER(Val(z0), Cons(Mul, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1))) -> c20(MEMBER(Val(z0), Cons(Plus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1))) -> c20(MEMBER(Val(z0), Cons(Minus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c20(MEMBER(Val(S(y0)), Cons(Val(S(y1)), y2))) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c20(MEMBER(Val(0), Cons(Val(S(y0)), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c20(MEMBER(Val(S(y0)), Cons(Val(0), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c20(MEMBER(Val(S(S(y0))), Cons(Val(S(S(y1))), y2))) MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (239) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x0)) -> c20(MEMBER(RPar, x0)) by MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(LPar, y0))) -> c20(MEMBER(RPar, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Div, y0))) -> c20(MEMBER(RPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Mul, y0))) -> c20(MEMBER(RPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Plus, y0))) -> c20(MEMBER(RPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Minus, y0))) -> c20(MEMBER(RPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Val(y0), y1))) -> c20(MEMBER(RPar, Cons(Val(y0), y1))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1))) -> c20(MEMBER(Val(z0), Cons(RPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1))) -> c20(MEMBER(Val(z0), Cons(LPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1))) -> c20(MEMBER(Val(z0), Cons(Div, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1))) -> c20(MEMBER(Val(z0), Cons(Mul, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1))) -> c20(MEMBER(Val(z0), Cons(Plus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1))) -> c20(MEMBER(Val(z0), Cons(Minus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c20(MEMBER(Val(S(y0)), Cons(Val(S(y1)), y2))) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c20(MEMBER(Val(0), Cons(Val(S(y0)), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c20(MEMBER(Val(S(y0)), Cons(Val(0), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c20(MEMBER(Val(S(S(y0))), Cons(Val(S(S(y1))), y2))) MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(LPar, y0))) -> c20(MEMBER(RPar, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Div, y0))) -> c20(MEMBER(RPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Mul, y0))) -> c20(MEMBER(RPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Plus, y0))) -> c20(MEMBER(RPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Minus, y0))) -> c20(MEMBER(RPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Val(y0), y1))) -> c20(MEMBER(RPar, Cons(Val(y0), y1))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2 ---------------------------------------- (241) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MEMBER(RPar, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, x2))) by MEMBER(RPar, Cons(LPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(LPar, y0)))) MEMBER(RPar, Cons(LPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Div, y0)))) MEMBER(RPar, Cons(LPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Mul, y0)))) MEMBER(RPar, Cons(LPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Plus, y0)))) MEMBER(RPar, Cons(LPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Minus, y0)))) MEMBER(RPar, Cons(LPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Val(y0), y1)))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: eqAlph(RPar, RPar) -> True eqAlph(RPar, LPar) -> False eqAlph(RPar, Div) -> False eqAlph(RPar, Mul) -> False eqAlph(RPar, Plus) -> False eqAlph(RPar, Minus) -> False eqAlph(RPar, Val(z0)) -> False eqAlph(LPar, RPar) -> False eqAlph(LPar, LPar) -> True eqAlph(LPar, Div) -> False eqAlph(LPar, Mul) -> False eqAlph(LPar, Plus) -> False eqAlph(LPar, Minus) -> False eqAlph(LPar, Val(z0)) -> False eqAlph(Div, RPar) -> False eqAlph(Div, LPar) -> False eqAlph(Div, Div) -> True eqAlph(Div, Mul) -> False eqAlph(Div, Plus) -> False eqAlph(Div, Minus) -> False eqAlph(Div, Val(z0)) -> False eqAlph(Mul, RPar) -> False eqAlph(Mul, LPar) -> False eqAlph(Mul, Div) -> False eqAlph(Mul, Mul) -> True eqAlph(Mul, Plus) -> False eqAlph(Mul, Minus) -> False eqAlph(Mul, Val(z0)) -> False eqAlph(Plus, RPar) -> False eqAlph(Plus, LPar) -> False eqAlph(Plus, Div) -> False eqAlph(Plus, Mul) -> False eqAlph(Plus, Plus) -> True eqAlph(Plus, Minus) -> False eqAlph(Plus, Val(z0)) -> False eqAlph(Minus, RPar) -> False eqAlph(Minus, LPar) -> False eqAlph(Minus, Div) -> False eqAlph(Minus, Mul) -> False eqAlph(Minus, Plus) -> False eqAlph(Minus, Minus) -> True eqAlph(Minus, Val(z0)) -> False eqAlph(Val(z0), RPar) -> False eqAlph(Val(z0), LPar) -> False eqAlph(Val(z0), Div) -> False eqAlph(Val(z0), Mul) -> False eqAlph(Val(z0), Plus) -> False eqAlph(Val(z0), Minus) -> False eqAlph(Val(z0), Val(z1)) -> !EQ(z1, z0) !EQ(S(z0), S(z1)) -> !EQ(z0, z1) !EQ(0, S(z0)) -> False !EQ(S(z0), 0) -> False !EQ(0, 0) -> True factor(Cons(RPar, z0)) -> z0 factor(Cons(Div, z0)) -> z0 factor(Cons(Mul, z0)) -> z0 factor(Cons(Plus, z0)) -> z0 factor(Cons(Minus, z0)) -> z0 factor(Cons(Val(z0), z1)) -> z1 factor(Cons(LPar, z0)) -> factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))) factor[Ite][True][Let](z0, Cons(RPar, z1)) -> factor[Ite][True][Let][Ite](True, z0, Cons(RPar, z1)) factor[Ite][True][Let](z0, Cons(LPar, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(LPar, z1)) factor[Ite][True][Let](z0, Cons(Div, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Div, z1)) factor[Ite][True][Let](z0, Cons(Mul, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Mul, z1)) factor[Ite][True][Let](z0, Cons(Plus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Plus, z1)) factor[Ite][True][Let](z0, Cons(Minus, z1)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Minus, z1)) factor[Ite][True][Let](z0, Cons(Val(z1), z2)) -> factor[Ite][True][Let][Ite](False, z0, Cons(Val(z1), z2)) factor[Ite][True][Let](z0, Nil) -> factor[Ite][True][Let][Ite](and(False, eqAlph(head(Nil), RPar)), z0, Nil) expr(z0) -> expr[Let](z0, term(z0)) expr[Let](z0, Cons(z1, z2)) -> expr[Let][Ite][False][Ite](member(z1, Cons(Plus, Cons(Minus, Nil))), z0, Cons(z1, z2)) expr[Let](z0, Nil) -> Nil term(z0) -> term[Let](z0, factor(z0)) term[Let](z0, Cons(z1, z2)) -> term[Let][Ite][False][Ite](member(z1, Cons(Mul, Cons(Div, Nil))), z0, Cons(z1, z2)) term[Let](z0, Nil) -> Nil member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](eqAlph(z1, z0), z0, Cons(z1, z2)) member(z0, Nil) -> False member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2) member[Ite][True][Ite](True, z0, z1) -> True Tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Val(S(z0)), Cons(Val(S(z1)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), Val(S(z0)), Cons(Val(S(z1)), x2)), EQALPH(Val(S(z1)), Val(S(z0)))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(LPar, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Div, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Mul, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x0)) -> c20(MEMBER(Minus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Plus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x0)) -> c20(MEMBER(RPar, x0)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x0)) -> c20(MEMBER(LPar, x0)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x0)) -> c20(MEMBER(Div, x0)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x0)) -> c20(MEMBER(Mul, x0)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x0)) -> c20(MEMBER(Plus, x0)) MEMBER[ITE][TRUE][ITE](False, Val(x0), Cons(Minus, x1)) -> c20(MEMBER(Val(x0), x1)) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(x0), x1)) -> c20(MEMBER(RPar, x1)) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(x0), x1)) -> c20(MEMBER(LPar, x1)) MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(x0), x1)) -> c20(MEMBER(Div, x1)) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(x0), x1)) -> c20(MEMBER(Mul, x1)) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(x0), x1)) -> c20(MEMBER(Plus, x1)) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(x0), x1)) -> c20(MEMBER(Minus, x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(S(x1)), x2)) -> c20(MEMBER(Val(S(x0)), x2)) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(x0)), x1)) -> c20(MEMBER(Val(0), x1)) MEMBER[ITE][TRUE][ITE](False, Val(S(x0)), Cons(Val(0), x1)) -> c20(MEMBER(Val(S(x0)), x1)) EXPR[LET](Cons(LPar, x0), Cons(RPar, z1)) -> c18(MEMBER(RPar, Cons(Plus, Cons(Minus, Nil)))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(Val(0), Cons(Val(S(z0)), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(Val(S(z0)), x2))) MEMBER(Val(S(z0)), Cons(Val(0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(z0)), Cons(Val(0), x2))) EXPR[LET](Cons(LPar, x0), Cons(LPar, z1)) -> c18(MEMBER(LPar, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Div, z1)) -> c18(MEMBER(Div, Cons(Plus, Cons(Minus, Nil)))) MEMBER(Val(S(S(y1))), Cons(Val(S(S(y0))), z2)) -> c30(EQALPH(Val(S(S(y0))), Val(S(S(y1))))) EXPR[LET](Cons(LPar, x0), Cons(Mul, z1)) -> c18(MEMBER(Mul, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Minus, z1)) -> c18(MEMBER(Minus, Cons(Plus, Cons(Minus, Nil)))) EXPR[LET](Cons(LPar, x0), Cons(Val(z1), z2)) -> c18(MEMBER(Val(z1), Cons(Plus, Cons(Minus, Nil)))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) TERM[LET](Cons(LPar, z0), Cons(RPar, z2)) -> c16(MEMBER(RPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(LPar, z2)) -> c16(MEMBER(LPar, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Div, z2)) -> c16(MEMBER(Div, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Plus, z2)) -> c16(MEMBER(Plus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Minus, z2)) -> c16(MEMBER(Minus, Cons(Mul, Cons(Div, Nil)))) TERM[LET](Cons(LPar, z0), Cons(Val(y0), z2)) -> c16(MEMBER(Val(y0), Cons(Mul, Cons(Div, Nil)))) !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c4(!EQ'(S(S(y0)), S(S(y1)))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(LPar, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(LPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(LPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(LPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(LPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(LPar, Cons(Val(y0), y1))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Div, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Div, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Div, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Div, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Div, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Div, Cons(Val(y0), y1))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Mul, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Mul, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Mul, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Mul, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Mul, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Mul, Cons(Val(y0), y1))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Plus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Plus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Plus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Plus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0))) -> c20(MEMBER(Plus, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Plus, Cons(Val(y0), y1))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0))) -> c20(MEMBER(Minus, Cons(RPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0))) -> c20(MEMBER(Minus, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0))) -> c20(MEMBER(Minus, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0))) -> c20(MEMBER(Minus, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0))) -> c20(MEMBER(Minus, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1))) -> c20(MEMBER(Minus, Cons(Val(y0), y1))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1))) -> c20(MEMBER(Val(z0), Cons(RPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1))) -> c20(MEMBER(Val(z0), Cons(LPar, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1))) -> c20(MEMBER(Val(z0), Cons(Div, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1))) -> c20(MEMBER(Val(z0), Cons(Mul, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1))) -> c20(MEMBER(Val(z0), Cons(Plus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1))) -> c20(MEMBER(Val(z0), Cons(Minus, y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c20(MEMBER(Val(S(y0)), Cons(Val(S(y1)), y2))) MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c20(MEMBER(Val(0), Cons(Val(S(y0)), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c20(MEMBER(Val(S(y0)), Cons(Val(0), y1))) MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c20(MEMBER(Val(S(S(y0))), Cons(Val(S(S(y1))), y2))) MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(LPar, y0))) -> c20(MEMBER(RPar, Cons(LPar, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Div, y0))) -> c20(MEMBER(RPar, Cons(Div, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Mul, y0))) -> c20(MEMBER(RPar, Cons(Mul, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Plus, y0))) -> c20(MEMBER(RPar, Cons(Plus, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Minus, y0))) -> c20(MEMBER(RPar, Cons(Minus, y0))) MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Val(y0), y1))) -> c20(MEMBER(RPar, Cons(Val(y0), y1))) MEMBER(RPar, Cons(LPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(LPar, y0)))) MEMBER(RPar, Cons(LPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Div, y0)))) MEMBER(RPar, Cons(LPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Mul, y0)))) MEMBER(RPar, Cons(LPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Plus, y0)))) MEMBER(RPar, Cons(LPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Minus, y0)))) MEMBER(RPar, Cons(LPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Val(y0), y1)))) S tuples: FACTOR(Cons(LPar, z0)) -> c29(EXPR(Cons(LPar, z0))) TERM(Cons(LPar, x0)) -> c85(FACTOR(Cons(LPar, x0))) TERM(Cons(LPar, z0)) -> c85(TERM[LET](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr(Cons(LPar, z0))))))), FACTOR(Cons(LPar, z0))) EXPR(Cons(LPar, z0)) -> c87(EXPR[LET](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor[Ite][True][Let](Cons(LPar, z0), expr[Let](Cons(LPar, z0), term[Let](Cons(LPar, z0), factor(Cons(LPar, z0))))))), TERM(Cons(LPar, z0))) K tuples: MEMBER(Mul, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(LPar, x2))) MEMBER(Plus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(LPar, x2))) MEMBER(Minus, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(LPar, x2))) MEMBER(Val(z0), Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(LPar, x2))) MEMBER(RPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Div, x2))) MEMBER(LPar, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Div, x2))) MEMBER(Mul, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Div, x2))) MEMBER(Plus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Div, x2))) MEMBER(Minus, Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Div, x2))) MEMBER(Val(z0), Cons(Div, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Div, x2))) MEMBER(RPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Mul, x2))) MEMBER(LPar, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Mul, x2))) MEMBER(Plus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Mul, x2))) MEMBER(Minus, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Mul, x2))) MEMBER(Val(z0), Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Mul, x2))) MEMBER(RPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Plus, x2))) MEMBER(LPar, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Plus, x2))) MEMBER(Mul, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Plus, x2))) MEMBER(Minus, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Plus, x2))) MEMBER(Val(z0), Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Plus, x2))) MEMBER(RPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Minus, x2))) MEMBER(LPar, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Minus, x2))) MEMBER(Mul, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Minus, x2))) MEMBER(Plus, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Minus, x2))) MEMBER(Val(z0), Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(Minus, x2))) MEMBER(RPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(Val(z0), x2))) MEMBER(LPar, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(Val(z0), x2))) MEMBER(Mul, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(Val(z0), x2))) MEMBER(Plus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(Val(z0), x2))) MEMBER(Minus, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(Val(z0), x2))) MEMBER(Div, Cons(LPar, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(LPar, x2))) MEMBER(Div, Cons(Mul, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Mul, x2))) MEMBER(Div, Cons(Plus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Plus, x2))) MEMBER(Div, Cons(Minus, x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Minus, x2))) MEMBER(Div, Cons(Val(z0), x2)) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(Val(z0), x2))) EQALPH(Val(S(S(y1))), Val(S(S(y0)))) -> c82(!EQ'(S(S(y0)), S(S(y1)))) MEMBER(LPar, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(RPar, y0)))) MEMBER(LPar, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Div, y0)))) MEMBER(LPar, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Mul, y0)))) MEMBER(LPar, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Plus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Minus, y0)))) MEMBER(LPar, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, LPar, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Div, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(RPar, y0)))) MEMBER(Div, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(LPar, y0)))) MEMBER(Div, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Mul, y0)))) MEMBER(Div, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Plus, y0)))) MEMBER(Div, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Minus, y0)))) MEMBER(Div, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Div, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Mul, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(RPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(LPar, y0)))) MEMBER(Mul, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Div, y0)))) MEMBER(Mul, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Plus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Minus, y0)))) MEMBER(Mul, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Mul, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Plus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Plus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Div, y0)))) MEMBER(Plus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Plus, Cons(RPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Minus, y0)))) MEMBER(Plus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Plus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Minus, Cons(RPar, Cons(RPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(RPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(LPar, y0)))) MEMBER(Minus, Cons(RPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Div, y0)))) MEMBER(Minus, Cons(RPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Mul, y0)))) MEMBER(Minus, Cons(RPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Plus, y0)))) MEMBER(Minus, Cons(RPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Minus, Cons(RPar, Cons(Val(y0), y1)))) MEMBER(Val(z0), Cons(RPar, Cons(RPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(RPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(LPar, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(LPar, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Div, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Div, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Mul, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Mul, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Plus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Plus, y1)))) MEMBER(Val(z0), Cons(RPar, Cons(Minus, y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(z0), Cons(RPar, Cons(Minus, y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(S(y1)), y2)))) MEMBER(Val(0), Cons(RPar, Cons(Val(S(y0)), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(0), Cons(RPar, Cons(Val(S(y0)), y1)))) MEMBER(Val(S(y0)), Cons(RPar, Cons(Val(0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(y0)), Cons(RPar, Cons(Val(0), y1)))) MEMBER(Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2))) -> c30(MEMBER[ITE][TRUE][ITE](False, Val(S(S(y0))), Cons(RPar, Cons(Val(S(S(y1))), y2)))) MEMBER(RPar, Cons(LPar, Cons(LPar, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(LPar, y0)))) MEMBER(RPar, Cons(LPar, Cons(Div, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Div, y0)))) MEMBER(RPar, Cons(LPar, Cons(Mul, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Mul, y0)))) MEMBER(RPar, Cons(LPar, Cons(Plus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Plus, y0)))) MEMBER(RPar, Cons(LPar, Cons(Minus, y0))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Minus, y0)))) MEMBER(RPar, Cons(LPar, Cons(Val(y0), y1))) -> c30(MEMBER[ITE][TRUE][ITE](False, RPar, Cons(LPar, Cons(Val(y0), y1)))) Defined Rule Symbols: eqAlph_2, !EQ_2, factor_1, factor[Ite][True][Let]_2, expr_1, expr[Let]_2, term_1, term[Let]_2, member_2, member[Ite][True][Ite]_3 Defined Pair Symbols: FACTOR_1, MEMBER_2, TERM_1, MEMBER[ITE][TRUE][ITE]_3, EXPR[LET]_2, EQALPH_2, TERM[LET]_2, !EQ'_2, EXPR_1 Compound Symbols: c29_1, c30_1, c30_2, c85_1, c20_1, c18_1, c82_1, c85_2, c16_1, c4_1, c87_2